6 ft
Find the area.
8 ft
8 ft
Remember: A = πr²
A = [?] ft²
Round to the nearest
hundredth.
Use 3.14 for T.

The temperature of the cup of water is approximately 180°F after 6 minutes.

Using the given formula, we can write:

T = Ta + (To - Ta) * e^(-kt)

where Ta = 73°F (the temperature of the room), To = 201°F (the initial temperature of the water), and T = 189°F (the temperature of the water after 3 minutes).

We can solve for the decay constant k as follows:

(T - Ta) / (To - Ta) = e^(-kt)

ln[(T - Ta) / (To - Ta)] = -kt

k = -ln[(T - Ta) / (To - Ta)] / t

Substituting the given values, we get:

k = -ln[(189°F - 73°F) / (201°F - 73°F)] / 3 minutes

k = -ln[116 / 128] / 3 minutes

k ≈ 0.0434 minutes^-1 (rounded to the nearest thousandth)

Now we can use this value of k to find the temperature of the water after 6 minutes:

T = Ta + (To - Ta) * e^(-kt)

T = 73°F + (201°F - 73°F) * e^(-0.0434 minutes^-1 * 6 minutes)

T ≈ 180°F (rounded to the nearest degree)

Therefore, the temperature of the cup of water is approximately 180°F after 6 minutes.

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The linear approximation of the given function f(x) is 40.5.

As f(X) = 4√(81+x) so by L(x) = f(a) + f'(a)(x-a) where a=0 and f(x)=4√(81+x) Then f'(x) = (1/2)(81+x)^(-1/2) So, f'(a) = (1/2)(81)^(-1/2) = 1/18.

Thus, L(x) = f(0) + f'(0)(x-0)

L(x) = 4√(81) + (1/18)x

L(x) = 36 + (1/18)x

Now we are asked to approximate the values 4√81.02 and 4√80.99 by using the linear approximation we found above. Let's solve the problems. Approximation of 4√81.02.L(x) = 36 + (1/18)x Now x=81.02. Thus, L(81.02) = 36 + (1/18)81.02L(81.02) = 36 + 4.5L(81.02) = 40.5.

Thus, the linear approximation of 4√81.02 is 40.5. Approximation of 4√80.99.L(x) = 36 + (1/18)x Now x=80.99. Thus, L(80.99) = 36 + (1/18)80.99L(80.99) = 36 + 4.49444...L(80.99) = 40.49444... ≈ 40.5. Thus, the linear approximation of 4√80.99 is approximately 40.5.

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Answer:

16x-20

Step-by-step explanation:

6(2x-3)-2(2x+1)

12x-18-4x+2

16x-20

Answer:

  10.309 in²/s

Step-by-step explanation:

Given A³ = 36πV² and V' = 18 in³/s, you want to know A' when A=153.24 in² and V=178.37 in³.

Using implicit differentiation, we have ...

  3A²·A' = 36π·2V·V'

  A' = (36π·2)/3·V/A²·V' = 24πV/A²·V'

  A' = 24π·(178.37 in²/(153.24 in²)²·18 in³/s

  A' ≈ 10.309 in²/s

The surface area is increasing at about 10.309 square inches per second.

__

Additional comment

There are at least a couple of ways a calculator can be used to find the rate of change. The first attachment shows evaluation of the expression we derived above. The second attachment shows the rate of change when the area is expressed as a function of the volume.

The result rounded to 5 significant figures is the same for both approaches.

Answer:

Parallelogram

Step-by-step explanation:

a quadrilateral (4 sides figure) whose opposite sides are parallel.  The opposite sides have the same length and opposite angles are equal.

Helping in the name of Jesus.

The answer of the question based on expressing the side of the base of a rectangular prism in terms of V and b the answer is the side of the base a is equal to the square root of the volume V divided by the height b.

Volume is measure of  amount of space that  three-dimensional object takes up. It is typically expressed in units of cubic length, like cubic meters, cubic feet, or cubic centimeters.

The formula for volume of many common geometric shapes can be calculated using mathematical formulas, while volumes of irregular shapes can be determined by measuring  amount of water they displace or by using advanced imaging techniques like CT scans.

The formula for  volume of  rectangular prism are:

V = length×width×height

Since the base of the prism is a square, the length and width are both equal to a. Therefore:

V = a × a × b

Simplifying this expression:

V = a²b

Dividing both sides by b:

a² = V/b

Taking the square root of both sides:

a = √(V/b)

Therefore, the side of the base a is equal to the square root of the volume V divided by the height b.

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Answer:

True

Step-by-step explanation:

To tell if you can make a triangle or not, it is necessary that the two shorter lengths added up must equal the longer length. In this case, the two shorter lengths are 5 and 5. 5 and 5 added up is 10 and 10>8 meaning that in this case, you can make a triangle

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Answer:

1

Step-by-step explanation:

Given values are:

x1 y1=(2,4)

x2 y2=( 4,6)

slop=(6-4)divide (4-2)=1

brainest please

thanks

Answer:

[tex]\frac{-1}{3}[/tex]

Step-by-step explanation:

Look at the red slope triangle that is drawn. If you start at the point on the left, you go down 1 unit and then to the right 3 units.  This would be represented by -1/3.  Down would be negative and going right would be positive.  The slope is the rise over the run.

Helping in the name of Jesus.

Answer: 12.84

Step-by-step explanation:

if x = 3.2 and y = 6.1  and Z = 0.2

then plug in the numbers

(3.2)(0.2) + (6.1)(2)

0.64 + 12.2 = 12.84

Any variable next to a number means multiplication.

if I was wrong lmk

The y intercept of line A is -2 times the y intercept of line B.

The line's intersection with the y-axis on a graph is known as the y-intercept. In every case, the associated x-coordinate is 0. This is accomplished by calculating the difference between the y- and x-coordinates and then dividing this difference.

standard form of equation of line:

y = mx + c

m is the slope and c is the y intercept.

Equation of line A:

3x + 2y = -6

Convert equation in standard form

2y = -3x -6

y = -3/2 x - 3

On comparing; y intercept = -3

Equation of line B. point taken (-3, 0) and (0,6)

y - y1 = m(x - x1)

slope m = (6 - 0)/(0 + 3) = 2

y - 0 = 2(x +3)

y = 2x  + 6

y intercept of line B is 6.

Thus,  y intercept  of line A is -2 times the y intercept of line B.

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The classmate's method is not correct because dividing the 1-week growth factor by 7 assumes that the growth rate is constant over each day of the week, which may not be the case so the correct 1-day correct growth factor is 1.036 and it is verified.

To calculate the correct 1-day growth factor, we can take the 7th root of the 1-week growth factor.

This is because if the height of the bamboo plant grows by a factor of x in one week, then the daily growth factor is the same as the weekly growth factor raised to the power of 1/7.

So the correct 1-day growth factor would be:

1-day growth factor = (1-week growth factor) raise to (1/7) = 1.27 raise to (1/7) ≈ 1.036

To verify that this is correct, we can raise the 1-day growth factor to the 7th power to obtain the 1-week growth factor:

1-week growth factor = (1-day growth factor)⁷ ≈ (1.036)⁷≈ 1.27

Therefore, the 1-day growth factor is approximately 1.036, and using this to find the 1-week growth factor yields approximately 1.27, which matches the given growth factor.

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The letters of the circled points based on their second coordinate values, from least to greatest R, A, T, I, and O.

A word that describes a way to compare quantities is RATIO.

In Mathematics, an ordered pair is sometimes referred to as a coordinate and it can be defined as a pair of two (2) elements that are commonly written in a fixed order within parentheses as (x, y), which represents the x-coordinate (abscissa) and the y-coordinate (ordinate) on the coordinate plane of any graph of a system of linear equations.

In this scenario, we can reasonably infer and logically deduce that the ordered pair representing the circled points are as follows:

Next, we would order the points by their second coordinate values, from least to greatest:

R = -4, A = 1, T = 2, I = 4, O = 5 ⇒ RATIO.

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Answer:

Step-by-step explanation:

a) To determine if the ball passes over the net, we need to find the height of the ball when it reaches x=12m (where the net is located).

y = -0.05x - 0.005x^2

y = -0.05(12) - 0.005(12)^2

y = -0.6 - 0.72

y = -1.32m

The ball reaches a height of -1.32m when it passes over the net, which is lower than the height of the net (0.9m). Therefore, the ball does pass over the net.

b) To determine if the serve is allowed, we need to find the horizontal distance where the ball lands and check if it is between the net and the service line (18m away from the origin).

We can find the distance the ball travels horizontally by finding the value of x when y = 0 (i.e., the height of the ball is 0 when it lands).

0 = -0.05x - 0.005x^2

0.005x^2 + 0.05x = 0

x(0.005x + 0.05) = 0

x = 0 (when the ball is hit) or x = -10 (when the ball lands)

The negative value of x indicates that the ball lands behind the server. Therefore, the serve is not allowed.

After solving the given equations we know that the value of x and y are -4 and -7 respectively.

A mathematical equation is a formula that uses the equals sign to represent the equality of two expressions.

Ax+By=C is the usual form for two-variable linear equations.

A standard form linear equation is, for instance, 2x+3y=5. When an equation is given in this format, finding both intercepts is rather simple (x and y).

This form is also highly useful when solving systems involving two linear equations.

So, we have the equations:

y=2x+1 ...(1)

-4x+y=9 ...(2)

Now, substitute y=2x+1 in equation (2):

-4x+y=9

-4x+2x+1=9

-2x=8

x=-4

Now, insert x=-4 in equation (1):

y=2x+1

y=2(-4)+1

y=-8+1

y=-7

Therefore, after solving the given equations we know that the value of x and y are -4 and -7 respectively.

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T(U) is a subspace of W Since V is a subspace of U, the zero vector of V, 0, is in U. As T is linear, [tex]T(0) = 0_w[/tex] , where [tex]0_w[/tex]  is the zero vector of W. Therefore, [tex]0_{w}[/tex] is in T(U).

It can be proved  Since U is a subspace of V, the zero vector of V, 0y, is in U. Since T is linear, [tex]T(0_{y}) = 0_{w}[/tex] , where [tex]0_{w}[/tex] is the zero vector of W. So[tex]0_{w}[/tex]  is in T(U).

To show that T(U) is a subspace of W, we also need to show that T(U) is closed under vector addition and scalar multiplication. Let u1, u2 be vectors in U, and let c be a scalar. Then we have: [tex]T(u1 + u2) = T(u1) + T(u2)[/tex]  (since T is linear)

[tex]T(cu1) = cT(u1)[/tex]  (since T is linear)

Since U is a subspace of V, we have [tex]u1 + u2[/tex] and [tex]cu1[/tex]  are also in U. Therefore, [tex]T(u1 + u2)[/tex]  and [tex]T(cu1)[/tex]  are both in T(U), which shows that T(U) is closed under vector addition and scalar multiplication.

Thus, T(U) is a subspace of W.

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The greatest number of 24 inch pieces the carpenter can cut from 6 boards of wood is 51.

A carpenter needs to cut 24-inch pieces of wood from a board that is 17 feet in length.

Therefore, the greatest number of 24 inches pieces the carpenter can cut from 6 of these boards of wood can be calculated as follows:

let's convert from feet to inches.

17 feet = 204 inches

1 board = 204 inches

6 board = ?

cross  multiply

length of 6 board = 204 × 6

length of 6 board = 1224 inches

Hence,

greatest number of 24 inch that can be cut from 6 boards = 1224 / 24

greatest number of 24 inch that can be cut from 6 boards = 51

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Answer:

If f(x)=x^3, evaluate f(x+h)-f(x)÷h, Where h* 0. Use your result to find the derivative of f(x) with respect to x. Differentiate with respect to x (x²-3x+5)(2x-7) .Find with respect to x the derivative of sinx ÷1– cosx​

Step-by-step explanation:

We are given f(x) = x^3. We need to find the value of (f(x+h) - f(x))/h.

f(x+h) = (x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3

Therefore, (f(x+h) - f(x))/h = [x^3 + 3x^2h + 3xh^2 + h^3 - x^3]/h

= 3x^2 + 3xh + h^2

Taking the limit of the above expression as h approaches 0, we get:

lim(h→0) [(f(x+h) - f(x))/h] = 3x^2

Therefore, the derivative of f(x) = x^3 with respect to x is 3x^2.

Next, we need to differentiate (x^2-3x+5)(2x-7) with respect to x.

Using the product rule, we get:

d/dx [(x^2-3x+5)(2x-7)] = (2x-7)(2x-3) + (x^2-3x+5)(2)

Simplifying, we get:

d/dx [(x^2-3x+5)(2x-7)] = 4x^2 - 20x + 11

Therefore, the derivative of (x^2-3x+5)(2x-7) with respect to x is 4x^2 - 20x + 11.

Finally, we need to find the derivative of sin(x)/(1-cos(x)) with respect to x.

Using the quotient rule, we get:

d/dx [sin(x)/(1-cos(x))] = [(1-cos(x))cos(x) - sin(x)(sin(x))]/(1-cos(x))^2

Simplifying, we get:

d/dx [sin(x)/(1-cos(x))] = cosec(x/2)^2

Therefore, the derivative of sin(x)/(1-cos(x)) with respect to x is cosec(x/2)^2.

The exponential function that models the quail population is:

P(t) = P0 * (1 + r)^t

where:

P0 is the initial population (385)

r is the annual growth rate (30% or 0.3)

t is the time in years

Substituting the values, we get:

P(t) = 385 * (1 + 0.3)^t

Simplifying:

P(t) = 385 * 1.3^t

To find the approximate population after 5 years, we substitute t = 5:

P(5) = 385 * 1.3^5

P(5) = 385 * 3.277

P(5) = 1262.45

Therefore, the approximate population after 5 years is 1262 quail

A system of linear inequalities that the graph represent include the following:

x ≥ 4.

y < -x - 2

y ≥ 3x + 3

y > x - 4

In Mathematics, the slope-intercept form of the equation of a straight line is given by this mathematical expression;

y = mx + c

Where:

At data points (-3, 0) and (0, 3), the slope of this line can be calculated as follows;

Slope = (3 - 0)/(0 + 3) = 3/3 = 1

y-intercept = 3.

Therefore, a linear inequality that models the line is given by:

y ≥ 3x + 3

At data points (-2, 0) and (0, -2), the slope of this line can be calculated as follows;

Slope = (-2 - 0)/(0 + 2) = -2/2 = -1

y-intercept = -2.

Therefore, a linear inequality that models the line is given by:

y < -x - 2

At data points (4, 0) and (0, -4), the slope of this line can be calculated as follows;

Slope = (-4 - 0)/(0 - 4) = -4/-4 = 1

y-intercept = -4.

Therefore, a linear inequality that models the line is given by:

y > x - 4

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Answer: D. Yes, the sum of the square of the legs is equal to the square of the hypotenuse.

Step-by-step explanation:

Pythagorean theorem is a^2 + b^2 = c^2

given side length of 6 and 8 and hypotenuse of 10

6^2 + 8^2 = 10^2

36 + 64 = 100

100 = 100

so yes, the sum of the square of the legs = the square of the hypotenuse.

Answer:

(2x - y)(3b + c + 2d)

Step-by-step explanation:

Factorize:

From the first two terms 6bx and (-3by) take out the common factor 3b.

From the third and fourth term 2cx and (-cy) take out the common factor c.

From the fifth and sixth term 4dx and (-2dy) take out the common factor 2d.

6bx - 3by  + 2cx - cy + 4dx - 2dy = 3b(2x - y) + c(2x - y) + 2d(2x - y)

                                                      = (2x - y)(3b + c + 2d)

RT arc length 3.31 units (rounded to two decimal places).

A circle is indeed a closed arc that extends outward from a set point known as the center, with each point on the curve being equally spaced from the center. [tex](x-h)2 + (y-k)2 = r2[/tex] is the equation for a circle having a (h, k) center and a r radius.

We may employ the following equation to determine that length of a circle's arc:

length of arc =[tex](central angle / 360 degrees) x 2πr[/tex]

Where the circle's radius, r, is.

According to the information provided in this problem, m/RST = 126, the arc RT's center angle is 126 degrees. We are also told that RS = 3 units, which indicates that the circle's radius is 3 units (since all radii of a circle have the same length).

Applying the formula with these values, we obtain:

length of arc [tex]RT = (126 / 360) x 2π(3) ≈ 3.31 units[/tex]

We obtain the final response by rounding to the closest hundredth:

RT arc length 3.31 units (rounded to two decimal places).

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Answer:

169 - 144p²

Step-by-step explanation:

(13 - 12p) × (13 + 12p)

each term in the second factor is multiplied by each term in the first factor

13(13 + 12p) - 12p(13 + 12p) ← distribute parenthesis

= 169 + 156p - 156p - 144p² ← collect like terms

= 169 - 144p²

The surface area of the object in the image is 80 cm². To find the surface area of the object in the image, we need to add up the areas of all the faces.

Most of the lateral faces are rectangular. Sometimes, it might even be a parallelogram. Here is the Prism Formula: A prism has a surface area of (2BaseArea) + Lateral Surface Area. Prism volume equals Base Area + Height.

The length of the rectangle is 4 cm and the width is 3 cm. Therefore, the area of the rectangular base is:

Area of rectangular base = length × width = 4 cm × 3 cm = 12 cm²

Area of each side = length × width = 5 cm × 3 cm = 15 cm²

Since there are four identical sides, the total area of all four sides is:

Total area of four sides = 4 × Area of each side = 4 × 15 cm² = 60 cm²

Area of each triangular face = 1/2 × base × height = 1/2 × 4 cm × 2 cm = 4 cm²

Since there are two triangular faces, the total area of both triangular faces is:

Total area of both triangular faces = 2 × Area of each triangular face = 2 × 4 cm² = 8 cm²

Now we can add up all the areas to get the total surface area:

Total surface area = Area of rectangular base + Total area of four sides + Total area of both triangular faces

= 12 cm² + 60 cm² + 8 cm²

= 80 cm²

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(a) The null hypothesis is that p = 0.5, and the alternative hypothesis is that p ≠ 0.5.

To conduct a hypothesis test, we need to define the null and alternative hypotheses. Let p be the proportion of US adults who believe raising the minimum wage will help the economy. The null hypothesis is that p = 0.5, and the alternative hypothesis is that p ≠ 0.5.

(b) The conditions for using a normal approximation to the binomial distribution are met since np = 1000(0.5) = 500 and n(1-p) = 1000(0.5) = 500 are both greater than 10.

(c) Using a significance level of α = 0.05, the test statistic is z = (0.42 - 0.5) / sqrt[(0.5)(0.5)/1000] = -2.38. The corresponding p-value is P(z < -2.38) = 0.017.

(d) Since the p-value is less than the significance level, we reject the null hypothesis. There is evidence to suggest that the proportion of US adults who believe raising the minimum wage will help the economy is not equal to 0.5. Therefore, we can conclude that a majority of US adults do not believe raising the minimum wage will help the economy.

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Answer:

Step-by-step explanation:

When a point is rotated, it must be rotated around a point.

See attachment for the image of the rotation about point K

We should note the following:

In order to construct triangles, you will need a protractor, a pair of compasses and a ruler. To draw the triangle, three properties must be taken into account: length, angle and shape

The given parameters are:

ΔEFG

The angle of rotation is

∅ = 180⁰

The above angle of rotation means that:

The translated triangle will be 180 degrees from ΔEFG about point K.

It also means that:

ΔEFG and ΔE'F'G' will be equidistant from point K

See attached image for ΔE'F'G'

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The probability that at least half of the 328 voters contacted in the telephone survey supported the current governor in the last election is 0.0532.

To calculate the probability that at least half of the 328 voters contacted in the telephone survey supported the current governor in the last election, we can use the binomial probability formula.

The formula is: P(x) = (ⁿCₓ) × pˣ × (1-p)⁽ⁿ⁻ˣ⁾

In this case, n = 328, p = 46.37%, and x = 164 (since half of 328 is 164).

Plugging in the numbers we get:

P(x) = ³²⁸C₁₆₄ × (0.4637)¹⁶⁴ × (0.5363)⁽³²⁸⁻¹⁶⁴⁾ = 0.0532

Therefore, the probability that at least half of the 328 voters contacted in the telephone survey supported the current governor in the last election is 0.0532.

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This type of growth model is often used to model revenue or sales of a product, as these types of processes often grow in a linear fashion.

Growth models, like those used in ForecastX, are typically used to model situations where a process grows at an exponential rate. This is a type of growth where the rate of increase is proportional to the size of the process, meaning that the process increases quickly over time.

This type of growth model is often used to model population growth, technological growth, and financial growth. In these cases, the growth model is usually used to make predictions about future growth in the process.  

In some cases, a growth model can be used to model a process that grows in a more or less constant rate. In this situation, the rate of increase is not proportional to the size of the process, and the rate of increase remains relatively stable over time. This type of growth model is often used to model natural resources, as these resources typically increase or decrease at a steady rate.

Finally, growth models can also be used to model a process that grows in a linear fashion. In this case, the rate of increase is constant and increases in a straight line over time.

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