Answer:
The simplified expression is sec^2a
Step-by-step explanation:
We can start by using the trigonometric identities:
cot^2 a + 1 = csc^2 a
tan^2 a + 1 = sec^2 a
Using these identities, we can rewrite the expression as:
(cos^2 a - cot^2 a)/(sin^2 a - tan^2 a)
= (cos^2 a - (csc^2 a - 1))/(sin^2 a - (sec^2 a - 1))
= (cos^2 a - csc^2 a + 1)/(sin^2 a - sec^2 a + 1)
Now we can use the identity:
sin^2 a + cos^2 a = 1
to rewrite the expression further:
= (1/sin^2 a - 1/sin^2 a cos^2 a)/(1/cos^2 a - 1/cos^2 a sin^2 a)
= (1 - cos^2 a)/(sin^2 a - sin^2 a cos^2 a)
= sin^2 a / sin^2 a (1 - cos^2 a)
= 1 / (1 - cos^2 a)
= sec^2 a
Therefore, the simplified expression is sec^2 a.
The ratio between two supplementary angle is 13:7. What are the measures of the angles?
Answer: The two angles are 117 degrees and 63 degrees.
Step-by-step explanation:
Supplementary angles are two angles whose sum is 180 degrees. Let the two angles be 13x and 7x, where x is a constant of proportionality.
We know that the sum of the angles is 180 degrees, so:
13x + 7x = 180
Combining like terms, we get:
20x = 180
Dividing both sides by 20, we get:
x = 9
So the measures of the angles are:
13x = 13(9) = 117 degrees
7x = 7(9) = 63 degrees
Therefore, the two angles are 117 degrees and 63 degrees.
the equations of motion of a two-degree-of-freedom system can be expressed in terms of the displacement of either of the two masses.true or false
The statement "the equations of motion of a two-degree-of-freedom system can be expressed in terms of the displacement of either of the two masses" is TRUE.
What are the equations of motion?The equations of motion refer to a set of mathematical equations that describe the behavior of a physical system over time. These equations define how the position, velocity, and acceleration of an object are related. The equations of motion are applicable to both single and multi-degree-of-freedom systems.
What is a two-degree-of-freedom system?A two-degree-of-freedom system is a physical system with two independent modes of motion or two degrees of freedom. It is defined by two generalized coordinates that completely define the system's state.
A two-degree-of-freedom system can be either linear or nonlinear, depending on the nature of the force. It is used in the study of structural dynamics, mechanical vibrations, and control engineering.
In a two-degree-of-freedom system, the equations of motion can be expressed in terms of the displacement of either of the two masses. The equations of motion are usually derived using Lagrange's equations, which are a set of equations that describe the dynamics of a mechanical system in terms of its energy. They are given as follows:
Where q₁ and q₂ are the generalized coordinates, m₁ and m₂ are the masses, k₁ and k₂ are the spring constants, and c₁ and c₂ are the damping coefficients.
These equations of motion are nonlinear and can be solved analytically or numerically using various techniques.
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The position vectors of points C and D are 2i + 37 and 7i + 5i respectively. Find the position vector of a point H which divides the line PQ in the ratio of 2: 3.
Answer:
Step-by-step explanation:
Let P be the point C with position vector 2i + 3j and Q be the point D with position vector 7i + 5j. We want to find the position vector of point H which divides PQ in the ratio 2:3.
Let the position vector of point H be r. Then we have:
PH/PQ = 2/3
(P - r)/(Q - r) = 2/3
Multiplying both sides by (Q - r), we get:
P - r = (2/3)(Q - r)
Expanding the brackets and rearranging, we get:
r = (2Q + 3P)/5
Substituting the position vectors of P and Q, we get:
r = (2(7i + 5j) + 3(2i + 3j))/5
= (14i + 10j + 6i + 9j)/5
= (20i + 19j)/5
= 4i + 3.8j
Therefore, the position vector of point H is 4i + 3.8j.
What is the ordered pair for y=3x-3
Answer:
Step-by-step explanation:
Choose three values for
x and substitute in to find the corresponding y values.(0,−3),(1,0),(2,3)
You can choose any of these three pairs or the equation y=3x-3
For what value of x is the figure a rectangle? Please help
Answer:
x = 8
Hope this helps!
Step-by-step explanation:
( 2x + 42 )° + ( 4x )° = 90° ( right angle )
6x + 42 = 90
6x = 90 - 42
6x = 48
x = 8
Complete the proof that △STU∼△WVU
So,To prove that △STU∼△WVU, we need to show that the two triangles are similar, which means that their corresponding angles are equal, and their corresponding sides are proportional.
What is triangle?A polygon with three sides and three angles is termed as triangle.
Triangles can be classified based on the length of their sides and the size of their angles, and they have a variety of properties and theorems associated with them in mathematics.
Angles: Since line VS is the bisecting line on line WT, angles VST and TSU are equal. Similarly, angles UVW and WVU are equal since U is the bisecting point on the line. Therefore, angle VUS is equal to angle WUV since they are vertically opposite angles.
Sides: We need to show that the corresponding sides are proportional.
Using the angle bisector theorem, we know that US/UT = VS/VT, which can be written as:
9/6 = VS/VT
Simplifying this, we get:
3/2 = VS/VT
Using the segment bisector theorem, we know that UW/WV = UT/TV, which can be written as:
6/4 = UT/TV
Simplifying this, we get:
3/2 = UT/TV
Since both ratios are equal to 3/2, we can say that:
VS/VT = UW/WV
Therefore, we can conclude that △STU∼△WVU by the angle-angle similarity theorem.
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Please help quick with this question.
Answer:
b = [tex]\frac{S-2la}{h+l}[/tex]
Step-by-step explanation:
S = bh + lb + 2la ( reversing the equation )
bh + lb + 2la = S ( subtract 2la from both sides )
bh + lb = S - 2la ← factor out b from each term on the left side
b(h + l) = S - 2la ← divide both sides by (h + l)
b = [tex]\frac{S-2la}{h+l}[/tex]
when solving arithmetic expressions, oracle 12c always resolves addition and subtraction operations first from left to right in the expression. true or false?
When solving arithmetic expressions, Oracle 12c resolves addition and subtraction operations first from left to right in the expression. This statement is true.
By following these rules, Oracle 12c is able to resolve arithmetic expressions accurately and efficiently. This is particularly important when dealing with large amounts of data, where the correct order of operations is critical to ensure that the correct result is obtained.
Oracle 12c is a database management system that is used to handle large amounts of data. It has a variety of features that enable it to manage and maintain data effectively.
Oracle 12c has a SQL engine that allows it to execute SQL statements and expressions to retrieve and manipulate data.
When performing arithmetic operations in Oracle 12c, the order of operations is critical. The order of operations is the set of rules that dictate the sequence in which arithmetic operations should be performed in a mathematical expression.
These rules are also known as the rules of precedence.In Oracle 12c, the rules of precedence dictate that arithmetic operations should be resolved from left to right in an expression. This means that addition and subtraction operations should be resolved before multiplication and division operations.
This is because addition and subtraction operations have a lower precedence than multiplication and division operations. As a result, Oracle 12c always resolves addition and subtraction operations first from left to right in an expression .This order of operations ensures that the correct result is obtained when performing arithmetic operations in Oracle 12c.
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enlarge triangle by scale factor 3 using the blue dot as the centre of enlargement
The new enlarged triangle has vertices at P4, P5, and P6, and is three times larger than the original triangle, with the blue dot as its center of enlargement.
What is a triangle?
A triangle is a geometric shape that consists of three straight sides and three angles. It is one of the most basic and fundamental shapes in geometry, and it is formed when three non-collinear points are connected by straight lines.
To enlarge a triangle by a scale factor of 3 using the blue dot as the center of enlargement, follow these steps:
Draw a line connecting the blue dot and each vertex of the triangle.
Mark a point on each line that is 3 times the distance from the blue dot as the corresponding vertex of the original triangle.
Connect the points on each line to form the new enlarged triangle.
Here is a step-by-step visual guide:
1. Draw lines connecting the blue dot to each vertex of the original triangle.
B
/\
/ \
/ \
P1 / \ P2
/ \
/ \
/____________\
A P3 C
2. Mark a point on each line that is 3 times the distance from the blue dot as the corresponding vertex of the original triangle.
B
/\
/ \
/ \
P1 / \ P2
/ \
/ P4 \
/____________\
A P3 C
Note that P4 is located 3 times the distance from the blue dot to vertex A along the line connecting them. Similarly, P5 and P6 are located 3 times the distance from the blue dot to vertices B and C respectively along their respective lines.
3. Connect the points on each line to form the new enlarged triangle.
P5
/\
/ \
/ \
P1 / \ P2
/ \
/ P4 \
/____________\
P3 P6 P7
Therefore, the new enlarged triangle has vertices at P4, P5, and P6, and is three times larger than the original triangle, with the blue dot as its center of enlargement.
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By how much percent will the area of circle will increase if the diameter increases by 40%?
The area of the circle will decrease by 51% if the diameter increases by 40%. Then the new diameter will be 1.4 times the original diameter.
Let's assume the original diameter is d and the original area of the circle is A.
Then, the new diameter will be 1.4d and the new area of the circle can be calculated as follows:
New area = pi * (New diameter/2)²= pi * (1.4d/2)² = pi * (0.7d)² = 0.49 * pi * d²
Therefore, the new area is 0.49 times the original area.
To calculate the percentage increase in area, we can use the following formula:
Percentage increase = (New value - Old value) / Old value * 100%
In this case, the old value is the original area A and the new value is the new area, which is 0.49*A. Therefore,
Percentage increase = (0.49A - A) / A * 100%
= -0.51A / A * 100%
= -51%
Therefore, the area of the circle will decrease by 51% if the diameter increases by 40%.
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Solve the triangle MNO (find the measure of ∠O and the lengths of sides MO and NO).
(I need help finding both side measures and angle measure please and thank you
Answer:
Step-by-step explanation:
m∠O = 90° - 34° = 56°
cos M = [tex]\frac{MN}{MO}[/tex] ⇒ MO = [tex]\frac{12}{cos34}[/tex] ≈ 14.5 cm
tan M = [tex]\frac{ON}{MN}[/tex] ⇒ ON = 12 × tan 34° ≈ 8.1 cm
Consider all four-digit numbers that can be created from the digits 0-9 where the first and last digits must be even and no digit can repeat. Assume that numbers can start with 0. What is the probability of choosing a random number that starts with 2 from this group? Enter a fraction or round your answer to 4 decimal places, if necessary
Answer:
Step-by-step explanation:
To find the probability of choosing a random number that starts with 2, we need to find two things:
1. The total number of valid four-digit numbers that can be created from the digits 0-9 where the first and last digits are even and no digit can repeat.
2. The number of valid four-digit numbers that start with 2 and meet the other criteria.
Let's begin by calculating the total number of valid four-digit numbers.
There are 5 even digits (0, 2, 4, 6, 8) to choose from for the first and last digits. For the second digit, there are 9 digits to choose from (since 0 cannot be repeated), for the third digit there are 8 digits to choose from (since we used one digit already and neither the first nor last digit can be repeated), and for the last digit there are only 4 choices left (since we used 5 for the first digit, and one more can’t be repeated). Therefore, the total number of valid four-digit numbers is:
5 * 9 * 8 * 4 = 1,440
Now we need to calculate the number of valid four-digit numbers that start with 2.
We already determined that there are 5 even digits to choose from for the last digit, leaving 8 digits for the second and 7 for the third digit (since the first digit is fixed). So the number of valid four-digit numbers that start with 2 is:
5 * 8 * 7 * 1 = 280
Therefore, the probability of choosing a random number that starts with 2 is:
280 / 1,440 = 0.1944 rounded to 4 decimal places.
So the answer to the problem is 97/500, which simplifies to 0.1940 when rounded to 4 decimal places.
In order for the parallelogram to be a
rectangle, x = [?].
Diagonal AC = 4x+68
Diagonal BD = 6x + 8
30 is the value of x in parallelogram .
What is a parallelogram?
With sides that are parallel to one another, a parallelogram is a two-dimensional geometric shape. The pair of parallel sides in this sort of polygon with four sides, also known as a quadrilateral, are of equal length. In a parallelogram, the sum of the adjacent angles equals 180 degrees.
4x+68 = 6x + 8
6x - 4x = 68 - 8
2x = 60
x = 30
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Im a trapezoid measuring 8cm, 10cm, 16 cm and 10cm on its sides. What is my Perimeter? 1-5 po lhat
Answer:
See Below.
Step-by-step explanation:
To find the perimeter of a trapezoid, you simply add up the lengths of all four sides.
In this case, the trapezoid has sides of 8 cm, 10 cm, 16 cm, and 10 cm.
Perimeter = 8 cm + 10 cm + 16 cm + 10 cm
Perimeter = 44 cm
Therefore, the perimeter of the trapezoid is 44 cm.
A movie theater is attracting customers with searchlights. One circular searchlight has a
radius of 2 feet. What is the searchlight's circumference?
Use 3.14 for л. If necessary, round your answer to the nearest hundredth.
The nearest hundredth, we get:
C ≈ 12.56 feet.
What is the value of 2r of a circle?Circle circumference (or perimeter) = 2R
where R denotes the circle's radius. 3.14 is the approximate (up to two decimal points) value of the mathematical constant. Again, Pi () is a special mathematical constant that represents the circumference to diameter ratio of any circle.
The circumference of a circle is calculated as follows:
C = 2πr
where C is the circumference, (pi) is a constant close to 3.14, and r is the radius of the circle.
When the given values are substituted, the following results are obtained:
C = 2(3.14)(2) \s= 12.56
We get the following when we round to the nearest hundredth:
C ≈ 12.56 feet.
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Help me with this question please
Answer:
x= 2√ 7
Step-by-step explanation:
We can solve it with applying the sin of 30°
Sin 30° = opposite leg / hypotenuse
Sin 30° = √ 7 / x
Being x unknown.
Now, being 30° a notable angle, we know that its sin is 1/2.
Sin 30° = 1/2
Replacing,
Sin 30° = √ 7 / x. (we clear x)
1/2 = √ 7 / x
x/2 = √ 7
x= 2√ 7
A triangle has area 100 square inches. It's dilated by a factor of k= 0.25.
Mai says, "The dilated triangle's area is 25 square inches."
Lin says, "The dilated triangle's area is 6.25 square inches."
1. For each student, decide whether you agree with their statement. If you
agree, explain why. If you disagree, explain what the student may have done to arrive at their answer.
2. Calculate the area of the image if the original triangle is dilated by each of these scale factors:
a. K = 9
b. K = 3/4
Lin's statement is correct. The incorrect answer is obtained due to inaccurate calculations of the new area of triangle.
What is dilation?A geometric modification called a dilation alters a figure's size. Each coordinate in the graphic is multiplied by a preset scale factor k to obtain it. The same element causes the figure to be stretched or contracted in all directions, creating a new figure that resembles the original but is smaller. Both two-dimensional (2D) and three-dimensional (3D) dilations are possible (3D). They are frequently employed in geometry to generate analogous figures and compute scale factors.
Mai's statement is incorrect. When a triangle is dilated by a factor of k, its area is multiplied by k².
When the given area of 100 square inches is dilated by a factor of 0.25 the new area is 6.25.
Hence, Lin's statement is correct. The incorrect answer is obtained due to inaccurate calculations of the new area.
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A surfboard is in the shape of a rectangle and semicircle. The perimeter is to be 4m. Find the maximum area of the surfboard correct to 2 places.
The maximum area of the surfboard correct to 2 places is 0.67 m².
Given that a surfboard is in the shape of a rectangle and a semicircle, and its perimeter is to be 4m. We need to find the maximum area of the surfboard, correct to 2 decimal places.
Let the radius of the semicircle be 'r' and the length and breadth of the rectangle be 'l' and 'b' respectively. Perimeter of the surfboard = [tex]4m => l + 2r + b + 2r = 4 => l + b = 4 - 4r[/tex] -----(1)
Area of surfboard = Area of rectangle + Area of semicircle Area of rectangle = l × b Area of semicircle = πr²/2 + 2r²/2 = (π + 2)r²/2Area of surfboard = l × b + (π + 2)r²/2 -----(2)
We have to maximize the area of the surfboard. So, we have to find the value of 'l', 'b', and 'r' such that the area of the surfboard is maximum .From equation (1), we have l + b = 4 - 4r => l = 4 - 4r - bWe will substitute this value of 'l' in equation (2)
Area of surfboard = l × b + (π + 2)r²/2 = (4 - 4r - b) × b + (π + 2)r²/2 = -2b² + (4 - 4r) b + (π + 2)r²/2Now, we have to maximize the area of the surfboard, that is, we need to find the maximum value of the above equation.
To find the maximum value of the equation, we can differentiate the above equation with respect to 'b' and equate it to zero. d(Area of surfboard)/db = -4b + 4 - 4r = 0 => b = 1 - r Substitute the value of 'b' in equation (1),
we get l = 3r - 3Now, we can substitute the values of 'l' and 'b' in the equation for the area of the surfboard.
Area of surfboard =
[tex]l × b + (π + 2)r²/2 = (3r - 3)(1 - r) + (π + 2)r²/2 = -r³ + (π/2 - 1)r² + 3r - 3[/tex]
[tex]-r³ + (π/2 - 1)r² + 3r - 3 = -0.6685 m² \\[/tex]
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Most people can roll their tongues, but many can’t. The ability to roll the tongue is genetically determined. Suppose we are interested in determining what proportion of students can roll their tongues. We test a simple random sample of 400 students and find that 317 can roll their tongues. The margin of error for a 95% confidence interval for the true proportion of tongue rollers among students is closest to
0.008.
0.02.
0.03.
0.04.
0.208.
The proportion of students who can roll their tongues will be estimated and the margin of error for a 95 percent confidence interval for the true proportion of tongue rollers among students will be determined. There were 317 tongue rollers out of a sample of 400 students.
As a result, the sample proportion is 317/400 = 0.7925.
We'll compute the margin of error next. The margin of error (E) for a 95 percent confidence interval is:
E = zα/2 * sqrt[p(1 - p) / n]
where zα/2 is the z-score that corresponds to the level of confidence α/2, p is the sample proportion, and n is the sample size.
E = 1.96 * sqrt[0.7925 * (1 - 0.7925) / 400]E
= 1.96 * sqrt[0.7925 * 0.2075 / 400]E
= 1.96 * sqrt(0.00040875)E
= 1.96 * 0.0202E
= 0.0395
The margin of error is approximately 0.04 or 4 percent. Hence, the correct option is 0.04.
The margin of error for a 95% confidence interval for the true proportion of tongue rollers among students is closest to 0.04
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A farmer has 20 boxes of eggs. There are 6 eggs in each box. Write, as a ratio, the number of eggs in two boxes to the total number of eggs. Give your answer in its simplest form.
Answer:
Step-by-step explanation:
number of eggs in 2 boxes = 12
Total number of eggs = 20 x 6 = 120
12:120
Simplify
2:20
Simplify
1:10
pls mark me brainiest
kernel composition rules 2 1 point possible (graded) let and be two vectors of the same dimension. use the the definition of kernels and the kernel composition rules from the video above to decide which of the following are kernels. (note that you can use feature vectors that are not polynomial.) (choose all those apply. )
a. 1
b. x.x’
c. 1+ x.x’
d. (1+ x.x’)^2
e. exp (x+x’), for x.x’ ER
f. min (x.x’) for x.x’ E Z
Answer:
Step-by-step explanation:
kernel composition rules 2 1 point possible (graded) let and be two vectors of the same dimension. use the the definition of kernels and the kernel composition rules from the video above to decide which of the following are kernels. (note that you can use feature vectors that are not polynomial.) (choose all those apply. )
how much does a typical water bed weigh? useful data: 1 cubic foot of water weighs 64.2 pounds and a typical water bed holds 28 cubic feet of water.
A typical water bed weighs around 1797.6 pounds. It depends upon the volume of water bed and the unit conversion of the weight and volume units.
What is the typical water bed weigh?A typical water bed holds 28 cubic feet of water.
The volume of the typical water bed and its weight can be calculated with the help of the quantities:
Identify the volume of water held by the water bed, which is 28 cubic feet.
Multiply the volume by the weight of 1 cubic foot of water, which is 64.2 pounds.
Perform the calculation: 28 cubic feet × 64.2 pounds per cubic foot = 1797.6 pounds.
Therefore, a typical water bed weighs approximately 1797.6 pounds when filled with water.
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The exponential 12 (3) 2x-12 has been converted to 12(k)*-6, what is the value of k?
Answer:
The solution set is (13,− 32). A quadratic equation of the form x 2= k can be solved by factoring with the following sequence of equivalent equations.
Step-by-step explanation:
9. Jess spins a pointer 25 times and finds an
experimental probability of the pointer landing
on 3 to be 25, or 16%. The theoretical probability
of the spinner landing on 3 is à, or 25%. Why
might there be a significant difference between
the theoretical and experimental probabilities?
The number of times the experiment is run affects the experimental probability.
What is Probability?Probability is the concept that describes the likelihood of an event occurring.
In real life, we frequently have to make predictions about how things will turn out.
We may be aware of the result of an occurrence or not.
When this occurs, we state that there is a possibility that the event will occur.
In general, probability has many excellent applications in games, commerce, and this newly growing area of artificial intelligence
The chance of an event can be calculated using the probability formula by only dividing the favourable number of possibilities by the total number of potential outcomes.
According to our question-
multiply that percentage by 25 to get the experimental probability percentage.
16%25
Hence, The number of times the experiment is run affects the experimental probability.
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PLEASE HELP I DONT HAVE TIME
Leo has a number of toy soldiers between 27 and 54. If he wants to group them four by four, there are none left, seven by seven, 6 remain, five by five, 3 remain. How many toy soldiers are there?
The answer is 48 but I need step by step explanation
The number of toy soldiers Leo has is 48.
How to calculate the number of toy soldiers Leo has?To solve this problem, we need to find a number that satisfies the three given conditions:
The number is between 27 and 54.
When the number is divided by 4, there is no remainder.
When the number is divided by 7, the remainder is 6.
When the number is divided by 5, the remainder is 3.
Let's start by finding a number that satisfies the first two conditions. We can do this by listing the multiples of 4 between 27 and 54:
28, 32, 36, 40, 44, 48, 52
Next, we need to find which of these numbers also satisfies the third condition, that is, leaves a remainder of 6 when divided by 7. We can go through each of these numbers and check:
28 divided by 7 leaves a remainder of 0.
32 divided by 7 leaves a remainder of 4.
36 divided by 7 leaves a remainder of 1.
40 divided by 7 leaves a remainder of 5.
44 divided by 7 leaves a remainder of 2.
48 divided by 7 leaves a remainder of 6.
52 divided by 7 leaves a remainder of 3.
So, the only number that satisfies the first three conditions is 48.
Finally, we need to check if 48 also satisfies the fourth condition, that is, leaves a remainder of 3 when divided by 5. Indeed, 48 divided by 5 leaves a remainder of 3.
Therefore, the number of toy soldiers Leo has is 48.
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One of the legs of a right triangle measures 4 cm and its hypotenuse measures 11 cm. Find the measure of the other leg. If necessary, round to the nearest tenth.
Answer: 10.2 cm
Step-by-step explanation:
We can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
Let's call the unknown length of the other leg "x". Then we can set up the following equation:
4^2 + x^2 = 11^2
Simplifying this equation, we get:
16 + x^2 = 121
Subtracting 16 from both sides, we get:
x^2 = 105
Taking the square root of both sides, we get:
x = sqrt(105)
x ≈ 10.2 cm (rounded to the nearest tenth)
Therefore, the measure of the other leg is approximately 10.2 cm.
9 of 109 of 10 items question a student prepares for a five-mile run so she eats a nutritious meal three hours before the run. which correctly illustrates the transformation of energy from eating the meal to the five-mile run?
The transformation of energy from eating the nutritious meal to the five-mile run involves the conversion of the nutrients from the meal into usable energy in the body. This energy is then converted into the kinetic energy necessary to move the student's body while running. The energy produced from the meal is broken down and used to fuel the muscles and other body systems necessary for the run.
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A random sample of 100 freshman showed 9 had satisfed the university mathematics requirement and a second random sample of 50 sophomores showed that 10 had satisfied the university mathematics requirement.
A) The relative risk of having satisfied the university mathematics requirement for sophomores as compared to freshmen is: 2.222
b) The increased risk of having satisfied the university mathematics requirement for sophomores as compard to freshmen is:
a) The relative risk of having satisfied the university mathematics requirement for sophomores as compared to freshmen is 2.222.
b) The increased risk of having satisfied the university mathematics requirement for sophomores as compared to freshmen is 122.2%.
Relative Risk is defined as the ratio of the risk of an event in the exposed group and the risk of the event in the unexposed group. The formula for relative risk is
RR = [a / (a + b)] / [c / (c + d)].
where, a = the number of individuals in the exposed group with the event,
b = the number of individuals in the exposed group without the event,
c = the number of individuals in the unexposed group with the event, and
d = the number of individuals in the unexposed group without the event.
Now, in this question, we have to calculate the relative risk and the increased risk for sophomores as compared to freshmen.
Given that,
Sample size of freshman = 100
Number of freshman who have satisfied the mathematics requirement = 9
Sample size of sophomores = 50
Number of sophomores who have satisfied the mathematics requirement = 10
a = 10, b = 40, c = 9, and d = 91
Relative risk for sophomores as compared to freshmen = (10 / (10 + 40)) / (9 / (9 + 91)) = 2.222
Therefore, the relative risk of having satisfied the university mathematics requirement for sophomores as compared to freshmen is 2.222.
Increased risk = (RR - 1) × 100% = (2.222 - 1) × 100% = 122.2%
Therefore, the increased risk of having satisfied the university mathematics requirement for sophomores as compared to freshmen is 122.2%.
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Let X denote the amount of time a book on two-hour reserve is actually checked out, and suppose the cdf is the following.0 , x<0f(x) = ((x^2)/4) , 0 <= x <= 21 , 2<= xUse the cdf to obtain the following. (If necessary, round your answer to four decimal places.)(a) P(X %u2264 1)(b) P(0.5 %u2264 X %u2264 1)(c) P(X > 1.5)(d) The median checkout duration [solve 0.5 = F(mew)](e) Use F'(x) to obtain the density function f(x)(f) Calculate E(X)(g) Calculate V(X) and %u03C3x(h) If the borrower is charged an amount h(X) = X2 when checkout duration is X, compute the expected charge E[h(X)].
By using CDF [tex]P(X \le 1)[/tex], [tex]P(0.5 \le X \le 1)=0.1875[/tex] , [tex]\mu= 1.4142[/tex], density function f(x) is [tex]0[/tex] [tex], E(X)=1 , V(X) = 1,[/tex] the expected charge [tex]E[h(X)]=2.[/tex]
(a) To obtain CDF
[tex]P(X \le 1)[/tex]
We have:
[tex]P(X \le 1) = F(1) = (1^2)/4 = 0.25[/tex]
(b) [tex]P(0.5 \le X \le1)[/tex]
We have:
[tex]P(0.5 \leX \le 1) \\ =F(1) - F(0.5) \\= (1^2)/4 - (0.5^2)/4 \\ =0.1875[/tex]
(c) [tex]P(X > 1.5)[/tex]
We have:
[tex]P(X > 1.5) \\= 1 - F(1.5) \\= 1 - (1.5^2)/4 \\= 0.1094[/tex]
(d) The median checkout duration [solve 0.5 = F(μ)]
To find the median, we need to solve the equation 0.5 = F(μ) for μ. This means we need to solve:
[tex]0.5 = (\mu^2)/4[/tex]
[tex]\mu^2= 2[/tex]
[tex]mu = \sqrt(2) \approx 1.4142[/tex]
(e) Use F'(x) to obtain the density function f(x)
We have:
[tex]f(x) = F'(x) \\= d/dx [(x^2)/4]\\ = x/2, 0 \le x \le 2[/tex]
[tex]f(x) = 0[/tex], otherwise
(f) Calculate [tex]E(X)[/tex]
We have:
[tex]E(X) = \int\ x f(x)dx[/tex] from 0 to 2
[tex]E(X) = \int\ x(x/2)dx[/tex] from 0 to 2
[tex]E(X) = [x^3/6][/tex] from 0 to 2
[tex]E(X) = (2^3/6)[/tex]
[tex]E(X) = 1[/tex]
(g) Calculate V(X) and σx
We have:
[tex]V(X) = E(X^2) - [E(X)]^2[/tex]
[tex]V(X) = \int\x^2f(x)dx[/tex] from 0 to 2 - [E(X)]²
[tex]V(X) = \int\x^2 (x/2)dx[/tex] from 0 to 2 - 1²
[tex]V(X) = [x^4/8][/tex] from 0 to 2 - 1
[tex]V(X) = (2^4/8) - (0^4/8) - 1[/tex]
[tex]V(X) = 1[/tex]
Therefore, [tex]\sigma x = \sqrt{(V(X))} = \sqrt{(1)} = 1.[/tex]
(h) If the borrower is charged an amount [tex]h(X) = X^2[/tex] when checkout duration is X, compute the expected charge E[h(X)]
We have:
[tex]E[h(X)] = \int\ h(x)f(x)dx[/tex] from 0 to 2
[tex]E[h(X)] = \int\x^2(x/2)dx[/tex] from 0 to 2
[tex]E[h(X)] = [x^4/8][/tex] from 0 to 2
[tex]E[h(X)] = (2^4/8) - (0^4/8)[/tex]
[tex]E[h(X)] = 2[/tex]
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50 POINTS!! Scientists measure a bacteria population and find that it is 10,000. Five days later,
they find that the population has doubled. Which function f could describe the
bacteria population d days after the scientists first measured it, assuming it grows
exponentially?
According to the solving function f could describe the bacteria population d days after the scientists first measured it, assuming it grows exponentially F(d) = 10000 - a[tex]^\frac{d}{5}[/tex].
What is function, exactly?A mathematical expression, rule, or law known as a function defines the relationship between two independent variables and a dependent variable (the dependent variable). In mathematics, functions are often utilised, and they are essential for creating physical connections in the sciences.
According to the given information:Substituting d = 5 and f(d)
Solution: fid) = 10000⋅ ad
{substitute d=5 and fid)=20000 into fid=10000. ady
10000- a⁵ =20000
a⁵ =2[tex]^\frac{1}{5}[/tex]
a = 2
F(d) = 10000 - a[tex]^\frac{d}{5}[/tex]
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