Answers
Therefore, the distance between the 90 degree angle and the hypotenuse is approximately 0.829 units.
What is triangle?A triangle is a two-dimensional geometric shape that is formed by three straight line segments that connect to form three angles. It is one of the most basic shapes in geometry and has a wide range of applications in mathematics, science, engineering, and everyday life. Triangles can be classified by the length of their sides (equilateral, isosceles, or scalene) and by the size of their angles (acute, right, or obtuse). The study of triangles is an important part of geometry, and their properties and relationships are used in many areas of mathematics and science.
Here,
1. To find HF, we can use the angle bisector theorem, which states that if a line bisects an angle of a triangle, then it divides the opposite side into two segments that are proportional to the adjacent sides. Let's denote the length of HF as x. Then, by the angle bisector theorem, we have:
JF/FH = JG/HG
Substituting the given values, we get:
15/x = 18/21
Simplifying and solving for x, we get:
x = 15 * 21 / 18
x = 17.5
Therefore, HF is 17.5 cm.
2. Let's denote the length of the hypotenuse as h and the length of the leg opposite the 18-unit perpendicular as a. We can then use the Pythagorean theorem to write:
h² = a² + 18²
We are told that the hypotenuse is divided into segments of length x and 6 units, so we can write:
h = x + 6
Substituting this expression into the first equation, we get:
(x + 6)² = a² + 18²
We are also told that the leg adjacent to the angle opposite the 4-unit segment is divided into segments of length 4 and (a - 4), so we can write:
a = 4 + (a - 4)
Simplifying this equation, we get:
a = a
Now we can substitute this expression for a into the previous equation and solve for x:
(x + 6)² = (4 + (a - 4))² + 18²
Expanding and simplifying, we get:
x² + 12x - 36 = 0
Using the quadratic formula, we get:
x = (-12 ± √(12² - 4(1)(-36))) / (2(1))
x = (-12 ± √(288)) / 2
x = -6 ± 6√(2)
Since the length of a segment cannot be negative, we take the positive root:
x = -6 + 6sqrt(2)
x ≈ 1.46
Therefore, the value of x is approximately 1.46 units.
3. Let's denote the length of the hypotenuse as h and the length of the leg adjacent to the angle opposite the 9-unit perpendicular as b. We can then use the Pythagorean theorem to write:
h² = b² + 9²
We are told that the hypotenuse is divided into segments of length x and 6 units, so we can write:
h = x + 6
Substituting this expression into the first equation, we get:
(x + 6)² = b² + 9²
Expanding and simplifying, we get:
x² + 12x - b² = 27
We also know that the length of the leg opposite the 9-unit perpendicular is:
a = √(h² - 9²)
= √((x + 6)² - 9²)
= √(x² + 12x + 27)
Now we can use the fact that the tangent of the angle opposite the 9-unit perpendicular is equal to the ratio of the lengths of the opposite and adjacent sides:
tan(θ) = a / b
Substituting the expressions for a and b, we get:
tan(θ) = √(x² + 12x + 27) / (x + 6)
We also know that the tangent of the angle theta is equal to the ratio of the length of the opposite side to the length of the adjacent side:
tan(θ) = 9 / b
Substituting the expression for b, we get:
tan(θ) = 9 / √(h² - 9²)
Substituting the expression for h, we get:
tan(θ) = 9 / √((x + 6)² - 9²)
Since the tangent function is the same for equal angles, we can set these two expressions for the tangent equal to each other:
√(x² + 12x + 27) / (x + 6) = 9 / √((x + 6)² - 9²)
Squaring both sides, we get:
(x² + 12x + 27) / (x + 6)² = 81 / ((x + 6)² - 81)
Cross-multiplying and simplifying, we get:
x⁴ + 36x³ + 297x² - 1458x - 2916 = 0
Using a numerical method such as the Newton-Raphson method or the bisection method, we can find the approximate solution to this equation:
x ≈ 9.449
Therefore, the value of x is approximately 9.449 units.
4. Let's denote the length of the hypotenuse as h and the length of the leg adjacent to the angle opposite the distance we want to find as b. We can use the Pythagorean theorem to write:
h² = b² + d²
We are told that the hypotenuse is divided into segments of length 9 and 4 units, so we can write:
h = 9 + 4 = 13
Substituting this expression into the first equation, we get:
13² = b² + d²
Simplifying and solving for d, we get:
d = √(13² - b²)
Now, we need to find the value of b. We know that the hypotenuse is divided into segments of length 9 and 4 units, so we can use similar triangles to write:
b / 4 = 9 / 13
Simplifying and solving for b, we get:
b = 36 / 13
Substituting this expression for b into the equation we found earlier for d, we get:
d = √(13² - (36/13)²)
Simplifying and finding a common denominator, we get:
d =√ (169*13 - 36²) / 13²
Simplifying further, we get:
d = √(169169 - 3636) / 169
Calculating this expression, we get:
d ≈ 0.829
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Related Questions
Suppose A and B are nxn matrices such that B and AB are both invertible. Prove that A is also invertible.
Hint: Show that A can be multiplied (on either side) by some other matrix or matrices to equal I
Answers
Matrix B and AB both are invertible implies that A is invertible as a matrix C such that AC = CA = I.
For matrix A to be invertible,
Show that there exists a matrix C such that AC = CA = I,
where I is the identity matrix.
Since B and AB are both invertible,
There exist matrices D and E such that ,
BD = DB = I
And ABE = EAB = I.
Multiplying both sides of the equation ABE = I by D on the left and E on the right, we get,
ADEBE = DE
Since BD = I, we can simplify this to,
ADE = DE
Multiplying both sides of this equation by B on the left and B^(-1) on the right, we get,
AD = DB^(-1)
Now, let C = DB^(-1).
Then we have,
AC
= ADB^(-1)
= ABEB^(-1)
= AI
= A
and
CA
= DB^(-1)A
= DB^(-1)ABE
= DI
= I
Therefore, A is invertible as a matrix C such that AC = CA = I.
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a cliff diver plunges from a height of 81 ft above the water surface. the distance the diver falls in t seconds is given by the function d(t)
Answers
(a) Therefore after t = 1.75 seconds the diver will hit the water.
(b) The velocity the diver hit the water is 56 ft/s.
From the given condition we have d(t) = 16t²
and the height is 49ft
(a) Now when the diver hit the water the equation become
16t² = 49
t² = 49/16
t = ±7/4
t = ±1.75
since time can not be negative so t = 1.75
Therefore after t = 1.75 seconds the diver will hit the water.
(b)
Now differentiating d(t) with respect to t we get
d'(t) = 32t
now putting t=7/4 we get
the velocity d'(7/4) = 32*7/4
d'(7/4) = 56ft/s
Therefore the velocity the diver hit the water is 56 ft/s.
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The complete question is :
A cliff diver plunges from a height of 49ft above the water surface. The distance the diver falls in t seconds is given by the function d(t)=16t²ft
(a) After how many seconds will the diver hit the water?
(b) With what velocity (in ft/s ) does the diver hit the water?
A camera has a list price of
$
459.99
before tax. If the sales tax rate is
7.25
%
,
find the total cost of the camera with sales tax included.
Round your answer to the nearest cent, if necessary.
Answers
In this case, we'll have to carry out several steps to find the solution.
Step 1:
Data:
camera:
list price (before tax) = $459.99
sales tax = 7.25%
Step 2:
percentage:
[tex]sales \ tax = 7.25\% = 7/100 = 0.07[/tex]
[tex]total \ cost = \$459.99 + \$459.99 \times (0.07) = \$459.99 + \$32.1993 = \$492.1893[/tex]
The answer is:
$492.19
Theresa wants to buy groceries that cost $2.99, $3.75, $1.09,
$4.50, $3.25, $2.58, $4.71, $5.19, $0.89, and $5.34. She has
$35. Estimate to see if she has enough money. Round up to
the nearest dollar.
Answers
Answer:
Total cost for groceries = ($3.75, $1.09,
$4.50, $3.25, $2.58, $4.71, $5.19, $0.89, and $5.34. add them all). = $ 31.3
the amount she paid= $ 35
balance =$ 3.7
therefore she have enough money
A cyclist rides off from rest, accelerating at a constant rate for 3 minutes until she reaches 40 kmh-1. She then maintains a constant speed for 4 minutes until reaching a hill. She slows down at a constant rate over one minute to 30 kmh-1. then continues at this rate for 10 minutes.
At the top of the hill she reduces her speed uniformly and is stationary 2 minutes later.
How far has the cyclist travelled?
Answers
Answer:
The cyclist has travelled a distance of 931.888 meters.
Hash 1 has an input data which is 2 characters long while Hash 2 has an input data which is 300000 characters long. Choose the correct option
Answers
The correct option on Hash 1 and Hash 2 is C. Hash 1 is designed to work with small input data, while Hash 2 is optimized for processing large input data.
What is the difference between Hash 1 and Hash 2 ?The main difference between Hash 1 and Hash 2 is the amount of input data they can process. Hash 1 can process input data that is 2 characters long, while Hash 2 can handle input data that is up to 300,000 characters long. This means that Hash 2 is better suited for processing larger datasets than Hash 1.
In terms of which hash function is better suited for different types of data, it depends on the specific application and the characteristics of the data being processed. For example, if the data being hashed is small and of low complexity, Hash 1 may be a good choice due to its speed and simplicity.
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The options for this question include:
Hash 1 is designed to work with large input data, while Hash 2 is optimized for processing small input data.Hash 1 is designed to work with small input data, while Hash 2 is optimized for processing small input data. Hash 1 is designed to work with small input data, while Hash 2 is optimized for processing large input data.Hash 1 is designed to work with large input data, while Hash 2 is optimized for processing large input data.in each of problems 9 and 10, use euler's method to find approximate values of the solution of the given initial value problem stemjock
Answers
The Euler's method was used to approximate the solution of two initial value problems at various time intervals with different step sizes. For problem, the solution is decreasing and converges to 1.
We will use the following formula for Euler's method:
y_{n+1} = y_n + h*f(t_n, y_n)
y' = 5 – 3sqrt(y), y(0) = 2
Using h = 0.1, we get:
t=0, y=2
t=0.1, y=1.738
t=0.2, y=1.508
t=0.3, y=1.303
t=0.4, y=1.119
t=0.5, y=0.953
Using h = 0.05, we get:
t=0, y=2
t=0.05, y=1.837
t=0.1, y=1.695
t=0.15, y=1.568
t=0.2, y=1.452
t=0.25, y=1.346
t=0.3, y=1.248
t=0.35, y=1.158
t=0.4, y=1.076
t=0.45, y=0.999
t=0.5, y=0.929
Using h = 0.025, we get:
t=0, y=2
t=0.025, y=1.861
t=0.05, y=1.737
t=0.075, y=1.622
t=0.1, y=1.516
t=0.125, y=1.418
t=0.15, y=1.328
t=0.175, y=1.246
t=0.2, y=1.17
t=0.225, y=1.101
t=0.25, y=1.038
t=0.275, y=0.98
t=0.3, y=0.927
t=0.325, y=0.878
t=0.35, y=0.833
t=0.375, y=0.791
t=0.4, y=0.753
t=0.425, y=0.718
t=0.45, y=0.685
t=0.475, y=0.655
t=0.5, y=0.627
Using h = 0.01, we get:
t=0, y=2
t=0.01, y=1.88
t=0.02, y=1.764
t=0.03, y=1.652
t=0.04, y=1.544
t=0.05, y=1.44
t=0.06, y=1.34
t=0.07, y=1.244
t=0.08, y=1.151
t=0.09, y=1.062
t=0.1, y=0.976
t=0.11, y=0.893
t=0.12, y=0.813
t=0.13, y=0.736
t=0.14, y=0.662
t=0.15, y=0.591
t=0.16, y=0.523
t=0.17, y=0.458
t=0.18, y=0
So, we can say that the step size had to be decreased to achieve more accurate approximations
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_____The given question is incomplete, the complete question is given below:
In each of Problems 9 and 10, use Euler's method to find approximate values of the solution of the given initial value problem at t = 0.5, 1, 1.5, 2, 2.5, and 3: a. With h=0.1. b. With h = 0.05. c. With h= 0.025. d. With h=0.01.
9. y' = 5 – 3 sqrt y, y(0) = 2
A large random sample of American students in seventh grade showed that
20
%
20%20, percent of them were reading below grade level.
Based on this data, which of the following conclusions are valid?
Choose 1 answer:
Choose 1 answer:
(Choice A) About
20
%
20%20, percent of all American students in seventh grade were reading below grade level.
A
About
20
%
20%20, percent of all American students in seventh grade were reading below grade level.
(Choice B)
20
%
20%20, percent of this sample was reading below grade level, but we cannot conclude anything about the population.
B
20
%
20%20, percent of this sample was reading below grade level, but we cannot conclude anything about the population.
(Choice C) About
20
%
20%20, percent of all American students were reading below grade level.
C
About
20
%
20%20, percent of all American students were reading below grade level.
Answers
The appropriate inference from the data is (B) Since [tex]20%[/tex] of this group read below grade level, we cannot draw any generalizations about the population. Thus, option B is correct.
What is the percent of the sample?A representative sample is a subset of data, often drawn from a wider population, that can show qualities that are similar.
Because the data produced contains more manageable, smaller representations of the larger group, representative sampling aids in the analysis of bigger groups.
Although the sample may be representative of seventh-grade American students, it is not necessarily representative of all seventh-graders or all American children. Hence, without additional data or research, we cannot extrapolate the sample's results to the overall population.
Therefore, 20%20, percent of all American students in seventh grade were reading below grade level.
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Find the average rate of change of the area of a circle withrespect to its radius r as r changes from2 to each of the following.(i) 2 to 3 (ii) 2 to 2.5 (iii) 2 to 2.1
Answers
The average rate of change is 5π; for r changing from 2 to 2.5, it is 2.5π, and for r changing from 2 to 2.1, it is 4.1π.
The area of a circle is given by the formula A = πr². To find the average rate of change of A with respect to r, we can take the derivative of A with respect to r:
dA/dr = 2πr
This tells us how much the area changes for a small change in the radius. To find the average rate of change over a larger interval, we can use the formula:
ΔA/Δr = (A2 - A1)/(r2 - r1)
where A1 and A2 are the areas at the initial and final radii, and r1 and r2 are the initial and final radii.
(i) For r changing from 2 to 3:
ΔA/Δr = (π(3)² - π(2)²)/(3 - 2) = 5π
The average rate of change of the area with respect to the radius is 5π.
(ii) For r changing from 2 to 2.5:
ΔA/Δr = (π(2.5)² - π(2)²/(2.5 - 2) = 2.5π
The average rate of change of the area with respect to the radius is 2.5π.
(iii) For r changing from 2 to 2.1:
ΔA/Δr = (π(2.1)² - π(2)²)/(2.1 - 2) = 4.1π
The average rate of change of the area with respect to the radius is 4.1π.
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Continuity find k (pre-calculus)!
Answers
so as we speak, the subfunctions are discontinued, the 1st goes close to 2 and who knows what happens it goes somewheres, the 2nd one makes it to 2.
we know that since the 2nd one makes to 2, to x = 2 that is, well, f(2) = kx, well, let's make f(2) for the 2nd one be equal to the 1st one then, if both they equate each other, that's where they meet, at x = 2.
[tex]f(x)= \begin{cases} k^2-24x,&x > 2\\\\ kx,&x\leqslant 2 \end{cases} \\\\[-0.35em] ~\dotfill\\\\ k^2-24x~~ = ~~kx\hspace{5em}\stackrel{\textit{now let's go to f(2)}}{k^2 - 24(2)~~ = ~~k(2)}\implies k^2-48=2k \\\\\\ k^2-2k-48=0\implies (k-8)(k+6)=0\implies \boxed{k= \begin{cases} 8\\ -6 \end{cases}}[/tex]
Assuming that the equation defines a differential function of x, find Dxy by implicit differentiation. 4)2xy-y2 = 1 5) xy + x + y = x2y2
Answers
For the equations 2xy - y^2 = 1 and xy + x + y = x^2y^2 using implicit differentiation the value Dxy is given by Dxy = (1 - 2xy + 3y^2)/(x - y)^3 and Dxy = (2y^2 - 2xy - 3y - 1)/(x - 2xy + 1)^3 respectively.
Equation 2xy - y^2 = 1,
Differentiate both sides of the equation with respect to x,
Treating y as function of x and then differentiate again with respect to x.
Using implicit differentiation,
First, differentiate both sides with respect to x,
2y + 2xy' - 2yy' = 0
Next, solve for y',
⇒2xy' - 2yy' = -2y
⇒y' (2x - 2y) = -2y
⇒y' = -y/(x - y)
Now, differentiate again with respect to x,
y''(x - y) - y'(2x - 2y) = y/(x - y)^2
Substitute the expression we obtained for y' in terms of y and x,
y''(x - y) - (-y/(x - y))(2x - 2y) = y/(x - y)^2
Simplify and solve for y'',
y''(x - y) + (2xy - 3y^2)/(x - y)^2 = 1/(x - y)^2
The expression for Dxy is,
Dxy = (1 - 2xy + 3y^2)/(x - y)^3
For the equation xy + x + y = x^2y^2,
Differentiate both sides of the equation with respect to x,
Using implicit differentiation,
First, differentiate both sides with respect to x,
⇒y + xy' + 1 + y' = 2xyy'
Solve for y',
⇒xy' - 2xyy' + y' = -y - 1
⇒y' (x - 2xy + 1) = -y - 1
⇒y' = -(y + 1)/(x - 2xy + 1)
Now, differentiate again with respect to x,
y''(x - 2xy + 1) - y'(2y - 2x y' + 1) = (y + 1)/(x - 2xy + 1)^2
Substitute the expression we obtained for y' in terms of y and x,
y''(x - 2xy + 1) - (-y - 1)/(x - 2xy + 1)^2 (2y - 2x y' + 1) = (y + 1)/(x - 2xy + 1)^2
Simplify and solve for y''
y''(x - 2xy + 1) - (2y^2 - 2xy - 2y)/(x - 2xy + 1)^2 = (y + 1)/(x - 2xy + 1)^2
The expression for Dxy is,
Dxy = (2y^2 - 2xy - 3y - 1)/(x - 2xy + 1)^3
Therefore , the value of Dxy using implicit differentiation for two different functions is equal to
Dxy = (1 - 2xy + 3y^2)/(x - y)^3 and Dxy = (2y^2 - 2xy - 3y - 1)/(x - 2xy + 1)^3
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60 percent of the songs Samir plays are 5 minutes long, 10 percent are 3 minutes long, and 30 percent are 2 minutes long. What is the average number of minutes per song ?
A. 1
B. 2
C. 3.9
D. 4.1
E. 4.5
Answers
Answer:
it's 3.9
Step-by-step explanation:
Assume Samir has total 100 songs and use combined mean formula
g let z denote the number of ones at the channel output. (z takes values 0, 1, ..., n.) specify the probability mass function
Answers
The probability mass function (PMF) for z, the number of ones at the channel output, can be expressed using the binomial distribution where p is the probability of transmitting a one and n is the total number of bits transmitted.
A probability mass function (PMF) is a function that assigns probabilities to each possible outcome in a discrete probability distribution. It describes the probability distribution of a discrete random variable, which takes on a finite or countably infinite number of possible values. The PMF is defined as the probability of each possible outcome, with the sum of all probabilities equal to 1. It is typically denoted as P(X = x), where X is the random variable and x is a possible value that it can take. The PMF is used to calculate various properties of the probability distribution, such as the expected value, variance, and higher moments.
The probability mass function (PMF) for z, the number of ones at the channel output, can be expressed using the binomial distribution formula:
[tex]$p(z) = \binom{n}{z} p^z (1-p)^{n-z}$[/tex]
where p is the probability of transmitting a one, n is the total number of bits transmitted, and [tex]$\binom{n}{z}$[/tex] is the binomial coefficient which counts the number of ways to choose z ones from n bits. The PMF specifies the probability of observing each possible value of z, ranging from 0 to n.
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PLEASEE HELP! DUE TONIGHT PLEASEE
find the area of a trapezoid
SHOW WORK:
Answers
Answer:
32 units square
Step-by-step explanation:
Area of trapezoid = 1/2 x h x (a +b) {a and b are parallel sides}
1st parallel side = 2 + 6 + 2
= 10 units
2nd parallel side = 6 units
height = 4 units
Area = 1/2 x 4 x (10+6)
= 2x16
= 32 units square
It is known that the area of a triangle can be calculated by multiplying the measure of the base by the measure of the height. Let the triangle measure 5m, 12m and 13m. Determine your area
Answers
The area of this triangle is 30 m².
What area?Area is a surface measure, that is, it is the amount of space that a geometric figure occupies on a flat surface.
To calculate the area of a triangle, we can use the formula:
Area = (base x height) / 2
In the case of the given triangle, we can choose the measure of 5m as the base and the measure of 12m as the height, since the height forms a right angle with the base and is perpendicular to it.
So, we have:
Area = (b*h)/2
Area = (5m * 12m) / 2
Area = 30m²
The ice cream above is going to melt.
When it does, will it fit in the cone or
will it overflow? Explain.
The spherical ice cream scoop and the
right cone have a radius of 3 cm.
The height of the çone is 13 cm.
Show all your work.
Answers
The ice cream scoop will fit inside the cone without overflowing, as shown by the volume comparison, which reveals that V ice cream > V cone.
what is cone ?A cone is a smooth-tapering, three-dimensional geometric shape with a flat base and a pointed tip or vertex. A cone is made up of a collection of line segments, half-lines, or lines that link the apex—the common point—to every point on a base that is in a plane other than the apex. The base can be any shape, but is most often a circle. Cones are frequently used in science and mathematics, as well as in commonplace items like ice cream cones, party hats, and traffic cones.
given
We need to compare their volumes to see if the ice cream scoop will fit inside the cone or spill out.
The quantity of the ice cream scoop could be determined by applying the following formula for the volume of a sphere:
[tex]V ice cream = (4/3)\pi r^3 \\= (4/3)\pi (3 cm)^3 \\= 113.1 cm^3[/tex]
[tex]V cone = (1/3)\pi r^2h \\= (1/3)\pi (3 cm)^2(13 cm) \\= 122.7 cm^3[/tex]
The ice cream scoop will fit inside the cone without overflowing, as shown by the volume comparison, which reveals that V ice cream > V cone.
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properties of the rectangle, rhombus, and square - practice determine if the following statements answers
Answers
1. The diagonals are equal. Rectangle
2. All sides are equal, and one angle is 60°. Rhombus
3. All sides are equal, and one angle is 90°. Square
4. It has all the properties of parallelogram, rectangle, and rhombus. Square
5. It is an equilateral parallelogram. Rhombus
A rectangle is a four-sided figure with two sets of parallel sides, with each side being a different length. The opposite sides of a rectangle are always equal in length, so the angles of a rectangle are all 90 degrees. A rectangle can also be referred to as a quadrilateral.
A rhombus is a four-sided figure with all sides the same length. The angles of a rhombus are not all 90 degrees, but the opposite sides of a rhombus are equal in length. A rhombus can also be referred to as a diamond.
A square is a four-sided figure with all sides being the same length and all angles being 90 degrees. A square can also be referred to as a regular quadrilateral.
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The complete question is:
Identify whether the following statements describe a rectangle, rhombus or square.
1. The diagonals are equal. ____________
2. All sides are equal, and one angle is 60°. ____________
3. All sides are equal, and one angle is 90°. ____________
4. It has all the properties of parallelogram, rectangle, and rhombus. ____________
5. It is an equilateral parallelogram. ____________
300 students attend Ridgewood Junior High School. 4% of students bring their lunch to school everyday. How many students brought their lunch to school on Thursday?
Answers
On Thursday, 12 students brought their lunch at school.
Define the term percentage?Using a number out of 100, a percentage is a technique to indicate a fraction or piece of a total. The word "percent" means "per hundred."
If 4% of the students bring their lunch to school every day, we can find the number of students who brought their lunch on Thursday by multiplying the total number of students by the percentage that brought their lunch:
Number of students who brought their lunch = (4/100) x 300
Number of students who brought their lunch = 12
Therefore, On Thursday, 12 students brought their lunch at school.
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A dietician is planning a snack package of fruit and nuts. Each ounce of fruit will supply zero units of protein, 3 units of carbohydrates, and 2 unit of fat, and will contain 40 calories. Each ounce of nuts will supply 4 units of protein, 2 unit of carbohydrate, and 4 units of fat, and will contain 50 calories. Every package must provide at least 4 units of protein, at least 11 units of carbohydrates, and no more than 16 units of fat. Find the number of ounces of fruit and number of ounces of nuts that will meet the requirement with the least number of calories. What is the least number of calories?
Let x be the ounces of fruit and y be the ounces of nuts. What is the objective function that must by minimized?
z = __x + __y
The dietician should use ___ ounce(s) of fruit and ___ ounce(s) of nuts. These amounts will have a total of ___calories.
(Type your answer in whole numbers)
Answers
The objective function that must be minimized is:
z = 40x + 50y
subject to the constraints:
0x + 4y ≥ 4 (protein constraint)
3x + 2y ≥ 11 (carbohydrate constraint)
2x + 4y ≤ 16 (fat constraint)
We want to find the number of ounces of fruit (x) and nuts (y) that will meet the requirement with the least number of calories.
Solving the system of inequalities, we get:
x = 2 ounces
y = 2 ounces
Therefore, the dietician should use 2 ounces of fruit and 2 ounces of nuts. These amounts will have a total of 180 calories (402 + 502).
Answer:
Step-by-step explanation:
Let's assume we need x ounces of fruit and y ounces of nuts to meet the requirements with the least number of calories. Then, the problem can be expressed as an optimization problem:
Minimize: 40x + 50y (since we want to minimize the number of calories) Subject to:
0x + 4y ≥ 4 (we need at least 4 units of protein)3x + 2y ≥ 11 (we need at least 11 units of carbohydrates)2x + 4y ≤ 16 (we cannot have more than 16 units of fat)
To solve this problem, we can use the simplex method. First, we convert the problem to standard form by introducing slack variables:
Minimize: 40x + 50y Subject to:
0x + 4y + s1 = 43x + 2y + s2 = 112x + 4y + s3 = 16
Now we can create the initial simplex tableau:
xys1s2s3RHSs1041004s23201011s32400116z-40-500000
We want to find the minimum value of z, so we need to choose the variable with the most negative coefficient in the bottom row as the entering variable. In this case, that is y. We then choose the variable with the smallest non-negative ratio between the right-hand side and the coefficient of the entering variable in its row as the leaving variable. In this case, that is s3, since 16/4 = 4 is the smallest non-negative ratio.
We then perform the pivot operation to eliminate the coefficient of y in the other rows:
x y s1s2s3RHSs1001-214y3/2101/2-1/24s2-1001-1/25z-100025-15200
We repeat this process until all the coefficients in the bottom row are non-negative. The final tableau is:
x
One year ago, JK Mfg. deposited $20,839 in an investment account for the purpose of buying new equipment four years from today. Today, it is adding another $22872 to this account. The company plans on making a final deposit of $20,217 to the account one year from today. How much will be available when it is ready to buy the equipment, assuming the company earns 10.91% APR on its invest funds?
Answers
Here is a step-by-step explanation for your problem:
Step 1: Calculate the amount of the first deposit after one year
First deposit: $20,839
Interest earned on first deposit: (20,839 x 10.91%) = $2,269.82
Total amount after one year: 20,839 + 2,269.82 = $23,108.82
Step 2: Calculate the amount of the second deposit after one year
Second deposit: $22,872
Interest earned on second deposit: (22,872 x 10.91%) = $2,511.33
Total amount after one year: 22,872 + 2,511.33 = $25,383.33
Step 3: Calculate the amount of the final deposit after one year
Final deposit: $20,217
Interest earned on final deposit: (20,217 x 10.91%) = $2,214.93
Total amount after one year: 20,217 + 2,214.93 = $22,432.93
Step 4: Calculate the total amount available after four years
Total amount available after four years = 23, 108.82 + 25,383.33 + 22,432.93 = $71,925.08
Allan painted the circular patch on his driveway. He used the formula below to calculate the area of the circular patch. The diameter of the circular patch was 20 meters. What was the area of the patch? Assume pi=3.14
Answers
Answer: 314 square meters
Step-by-step explanation:
The formula for the area of a circle is given by A = πr^2, where r is the radius of the circle. Since the diameter of the circular patch is given as 20 meters, the radius would be half of that or 10 meters.
So, using the formula, we can calculate the area of the circular patch as follows:
A = πr^2
A = π(10)^2
A = 3.14(100)
A = 314 square meters
Therefore, the area of the circular patch is 314 square meters.
A bicycle wheel is 63m in diameter. how many metres does the bicycle travel for 100 revolutions of the wheel. (pie=²²/⁷
Answers
Answer:
19782m
Step-by-step explanation:
1 revolution = circumference
circumference = π * diameter
π = 3.1416
Then
circumference = 3.1416 * 63
= 197.92m
1 revolution = 197.82m
100 revolutions = 100*197.82m
= 19782m
Answer:
19.8 km
Step-by-step explanation:
To find:-
The distance travelled in 100 revolutions .Answer:-
We are here given that,
diameter = 63mWe can first find the circumference of the wheel using the formula,
[tex]:\implies \sf C = 2\pi r \\[/tex]
Here radius will be 63/2 as radius is half of diameter. So on substituting the respective values, we have;
[tex]:\implies \sf C = 2\times \dfrac{22}{7}\times \dfrac{63}{2} \ m \\[/tex]
[tex]:\implies \sf C = 198\ m \\[/tex]
Now in one revolution , the cycle will cover a distance of 198m . So in 100 revolutions it will cover,
[tex]:\implies \sf Distance= 198(100)m\\[/tex]
[tex]:\implies \sf Distance = 19800 m \\[/tex]
[tex]:\implies \sf Distance = 19.8 \ km\\[/tex]
Hence the bicycle would cover 19.8 km in 100 revolutions.
se spherical coordinates to evaluate the triple integral where is the region bounded by the spheres and .
Answers
The value of the triple integral[tex]\int \int\int _{E } \frac{e^{-(x^2+y^2+z^2)}}{\sqrt{(x^2+y^2+z^2}}\sqrt{dV}[/tex] by using spherical coordinates [tex]2\pi(e^{-1}-e^{-9})[/tex].
Given that the triple integral is-
[tex]\int \int\int _{E } \frac{e^{-(x^2+y^2+z^2)}}{\sqrt{(x^2+y^2+z^2}}\sqrt{dV}[/tex]
E is the region bounded by the spheres which are,
[tex]x^2+y^2+z^2=1\\\\x^2+y^2+z^2=9[/tex]
In spherical coordinates we have,
x = r cosθ sin ∅
y = r sinθ sin∅
z = r cos∅
dV = r²sin∅ dr dθ d∅
E contains two spheres of radius 1 and 3 () respectively, the bounds will be like this,
1 ≤ r ≤ 3
0 ≤ θ ≤ 2π
0 ≤ ∅ ≤ π
Then
[tex]\int \int\int _{E } \frac{e^{-(x^2+y^2+z^2)}}{\sqrt{(x^2+y^2+z^2}}\sqrt{dV}[/tex]
[tex]\int\int\int _{E} \frac{e^{-r^2}}{r}r^2Sin\phi drd\phi d\theta\\\\2\pi \int_{0}^{\pi} \int_1^3 re^{-r^2} dr d\phi\\\\2\pi \int_1^3 re^{-r^2} dr\\\\2\pi(e^{-1}-e^{-9})[/tex]
The complete question is-
Use spherical coordinates to evaluate the triple integral ∭ee−(x2 y2 z2)x2 y2 z2−−−−−−−−−−√dv, where e is the region bounded by the spheres x2 y2 z2=1 and x2 y2 z2=9.
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What is the meaning of "permutations that preserve distances"?
Answers
Answer: Permutations that preserve distances are also known as isometries or distance-preserving transformations.
Step-by-step explanation:
Permutations that preserve distances refer to a type of mathematical transformation that preserves the distances between pairs of points in a geometric space. In other words, if you have a set of points arranged in a particular way and you apply a permutation that preserves distances, the resulting arrangement of points will have the same distances between each pair of points as the original arrangement. This type of permutation is important in geometry and can be used to study properties of geometric objects such as polyhedra, graphs, and other structures. Permutations that preserve distances are also known as isometries or distance-preserving transformations.
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A parallelogram is a type of quadrilateral.
A) False
B) True
Answers
Answer:
True
Step-by-step explanation:
A parallelogram has four sides so it's a quadrilateral
How many fractions between and inclusive can be written with a
denominator of 15?
Answers
The number of fractions between 0 and 1 (inclusive) with a denominator of 15 can be found using the formula (n-1)/n, where n is the denominator.
So, to answer your question, we can use the formula and plug in 15 for the value of n:
(15-1)/15 = 14/15
Therefore, there are 14 fractions between 0 and 1 (inclusive) with a denominator of 15.
evaluate 53 - 3^2 X 2
Answers
[tex]53 - 3^2 * 3 = 35[/tex]
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A commuter railway has 800 passengers per day and charges each one two dollars per day. For each 4 cents that the fare is increased, 5 fewer people will go by train.
What is the greatest profit that can be earned?
Greatest profit = $_____
Answers
Answer:
Step-by-step explanation:
To find the greatest profit, we need to determine the fare that will maximize revenue, while also considering the decrease in ridership due to the fare increase.
Let's assume the initial fare is $2, and the number of passengers is 800 per day. So, the initial revenue is:
$2 x 800 = $1600 per day
Now, let's say we increase the fare by 4 cents to $2.04. According to the problem, for each 4 cents increase in fare, there will be 5 fewer passengers. So, the number of passengers will decrease to:
800 - (5 x 4) = 780 passengers per day
The new revenue at this fare will be:
$2.04 x 780 = $1591.20 per day
By increasing the fare, the revenue decreased. This means that we may have increased the fare too much. Let's try another fare.
If we increase the fare by 2 cents to $2.02, the number of passengers will decrease by:
800 - (5 x 2) = 790 passengers per day
The new revenue at this fare will be:
$2.02 x 790 = $1595.80 per day
This is more revenue than the initial fare of $2 per person. Let's continue this process:
If we increase the fare by another 2 cents to $2.04, the number of passengers will decrease by:
790 - (5 x 2) = 780 passengers per day
The new revenue at this fare will be:
$2.04 x 780 = $1591.20 per day
This is less revenue than the $2.02 fare, so we can stop here.
Therefore, the greatest profit can be earned by charging $2.02 per person per day, and the maximum revenue will be:
$2.02 x 790 = $1595.80 per day
This is a bit less than the initial daily revenue of $1600, but it is the most revenue we can get by increasing the fare without causing a significant reduction in ridership.
Answer:
$2205
Step-by-step explanation:
You want the greatest profit that can be earned by a commuter railway that has 800 passengers per day at a fare of $2, and 5 fewer for each 4¢ increase in the fare.
Ridership functionThe number of riders (q) as a function of price (p) can be described by ...
q = 800 -5(p -2)/0.04
q = 1050 -125p . . . . . . . simplified
Revenue functionThe daily revenue is the product of price and the number of riders who pay that price.
r = pq
r = p(1050 -125p)
r = 125p(8.40 -p)
Maximum revenueThis function describes a parabola that opens downward. It has zeros at p=0 and p=8.40. The vertex of the parabola is on the line of symmetry, halfway between the zeros:
pmax = (0 +8.40)/2 = 4.20
The maximum revenue is ...
r(4.20) = 125·4.20(8.40 -4.20) = 125(4.20²) = 2205
The maximum revenue that can be earned is $2205.
__
Additional comment
The ridership at that fare is 125(4.20) = 525.
Profit is the difference between revenue and cost. Here, we have no information about the cost function, so we cannot predict the maximum profit. The question seems to assume that profit is equal to revenue.
Find the area of the parallelogram. Round to the nearest hundredth if necessary.
Answers
Answer:
Step-by-step explanation:
5m(4m) = 20m^2
The toll T charged for driving on a certain stretch of a toll road is $5 except during rush hours (between 7 AM and 10 AM and between 4 PM and 7 PM) when the toll is $7.
Answers
The toll calculation for driving on a certain stretch of a toll road is $5 except during rush hours when the toll is $7, depending on the time the driver uses the toll road.
To compute the toll for driving on the toll road during non-rush hours, simply add $5 to the total. During rush hour, however, the toll is $7.
To compute the toll for driving during rush hour, you must first determine when the driver intends to utilize the toll road. If the period is between 7 AM and 10 AM or 4 PM and 7 PM, the toll is $7.
For instance, if a vehicle expects to use the toll road at 8 a.m., the toll is $7. If the vehicle intends to use the toll road at 2 p.m., the toll is $5.
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