Spheres - ACT Math

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Question

If a sphere has a volume of $36\pi$ , what is its diameter?

Answer

1. Use the volume to find the radius:

$Volume=\frac{4}{3}\pi r^{3}$

$36\pi = \frac{4}{3}\pi r^{3}$

$\frac{3}{4}(36\pi) = (\frac{4}{3}\pi r^{3})\frac{3}{4}$

$27=r^{3}$

2. Use the radius to find the diameter:

$d=2r=2\cdot 3=6$

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Question

A sphere has a volume of $36\pi$ . What is its diameter?

Answer

This question relies on knowledge of the formula for volume of a sphere, which is as follows: $V=\frac{4}{3}\pi r^{3}}$

In this equation, we have two variables, and . Additionally, we know that $V=36\pi$ and is unknown. You can begin by rearranging the volume equation so it is solved for , then plug in and solve for :

Rearranged form:

$r=\sqrt[3]{\frac{3}{4\pi}V}$

Plug in $36\pi$ for V

$r=\sqrt[3]{\frac{3}{4\pi}36\pi}$

Simplify the part under the cubed root

Cancel the $\pi$ 's since they are in the numerator and denominator.

Simplify the fraction and the :

$36*\frac{3}{4}=27$

Thus we are left with

$r=\sqrt[3]{27}$

Then, either use your calculator and enter $27^{\frac{1}{3}}$ Or recall that $3^{3}=27$ in order to find that .

We're almost there, but we need to go a step further. Dodge the trap answer "" and carry on. Read the question carefully to see that we need the diameter, not the radius.

So

is our final answer.

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Question

A spherical plastic ball has a diameter of . What is the volume of the ball to the nearest cubic inch?

Answer

To answer this question, we must calculate the volume of the ball using the equation for the volume of a sphere. The equation for the volume of a sphere is four-thirds multiplied by pi, which is then multiplied by the radius cubed. The equation can be written like this:

$V = \frac{4}{3}\pi r^{3}$

We are given the diameter of the sphere in the problem, which is . To get the radius from the diameter, we divide the diameter by . So, for this data:

$radius=\frac{diameter}{2}=\frac{4}{2}=2$

We can then plug our newly found radius of two into the equation to find the volume. For this data:

$Volume = \frac{4}{3}\pi r^{3}=\frac{4}{3}\pi \cdot (2)^{3} = \frac{4}{3}\pi\cdot 8$

We then multiply $\frac{4}{3}$ by .

$\frac{4}{3}\pi\cdot 8=\frac{32}{3}\pi$

We finally substitute 3.14 for pi and multiply again to get our answer.

$\frac{32}{3}\pi = 33.5$

The question asked us to round to the nearest whole cubic inch. To do this, we round a number up one place if the last digit is a 5, 6, 7, 8, or 9, and we round it down if the last digit is a 1, 2, 3, or 4. Therefore:

$33.5\rightarrow34$

Therefore our answer is .

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Question

A boulder breaks free on a slope and rolls downhill. It rolls for complete revolutions before grinding to a halt. If the boulder has a volume of cubic feet, how far in feet did the boulder roll? (Assume the boulder doesn't lose mass to friction). Round $\pi$ to 3 significant digits. Round your final answer to the nearest integer.

Answer

The formula for the volume of a sphere is:

$V = \frac{4}{3}\pi r^3$

To figure out how far the sphere rolled, we need to know the circumference, so we must first figure out radius. Solve the formula for volume in terms of radius:

$1436 = \frac{4}{3}\pi r^3$

$\frac{1077}{\pi} =r^3$

$r \approx 6.999932$

Since the answer asks us to round to the nearest integer, we are safe to round to at this point.

To find circumference, we now apply our circumference formula:

$C = 2r\pi = 14\pi$

If our boulder rolled times, it covered that many times its own circumference.

$355\cdot 14\pi = 4970\pi \approx 15606$

Thus, our boulder rolled for

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Question

Find the diameter of a sphere whose radius is .

Answer

To solve, simply remember that diameter is twice the radius. Don't be fooled when the radius is an algebraic expression and incorporates the arbitrary constant . Thus,

$\textup{diameter}=2r=2*2d=4d$

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Question

The surface area of a sphere is $64\pi$ feet. What is the radius?

Answer

Solve the equaiton for the surface area of a sphere for the radius and plug in the values:

$SA=4{\pi}r^2\rightarrow r=\sqrt{\frac{SA}{4\pi}}= \sqrt{\frac{64\pi}{4\pi}}=\sqrt16=4$

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Question

What is the radius of a sphere with a volume of $2304\pi$ ? Round to the nearest hundredth.

Answer

Recall that the equation for the volume of a sphere is:

$V=\frac{4}{3}\pi r^3$

For our data, we know:

$2304\pi=\frac{4}{3}\pi r^3$

Solve for . First, multiply both sides by $\frac{3}{4}$ :

$\frac{3}{4}2304\pi=\pi r^3$

$1728\pi=\pi r^3$

Now, divide out the $\pi$ :

Using your calculator, you can solve for . Remember, if need be, you can raise to the power of $\frac{1}{3}$ if your calculator does not have a variable-root button.

This gives you:

If you get something like , just round up. This is a rounding issue with some calculators.

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Question

The volume of a sphere is $972\pi:cm^3$ . What is the diameter of the sphere? Round to the nearest hundredth.

Answer

Recall that the equation for the volume of a sphere is:

$V=\frac{4}{3}\pi r^3$

For our data, we know:

$972\pi = \frac{4}{3}\pi r^3$

Solve for . Begin by dividing out the $\pi$ from both sides:

$972 = \frac{4}{3} r^3$

Next, multiply both sides by $\frac{3}{4}$ :

$\frac{3}{4} * 972 =r^3$

Using your calculator, solve for . Recall that you can always use the $\frac{1}{3}$ power if you don't have a variable-root button.

You should get:

If you get , just round up to . This is a general rounding problem with calculators. Since you are looking for the diameter, you must double this to .

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Question

What is the radius of a sphere with a surface area of $169\pi$ ? Round to the nearest hundredth.

Answer

Recall that the surface area of a sphere is found by the equation:

$SA = 4\pi r^2$

For our data, this means:

$169\pi = 4\pi r^2$

Solve for . First, divide by $4\pi$ :

Take the square root of both sides:

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Question

What is the radius of a sphere with a volume of $36\prod \textup{ units}^{3}$ ?

Answer

Given the volume of the sphere, $36\prod \textup{units}^{3}$ , you need to use the formula for volume of a sphere $\left ( \left ( \frac{4}{3} \right )\cdot \prod \cdot r^{3}\right )$ and work backwards to find the radius. I would multiply both sides by $\frac{3}{4}$ to get rid of the $\frac{4}{3}$ in the formula. You then have $27\prod=\prod r^{3}$ . Next, divide both sides by $\prod$ so that all vyou have left is $27=r^{3}$ . Finally take the cube root of , to get units for the radius.

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Question

A cube with sides of is circumscribed by a sphere, such that all eight vertices of the cube are tangent to the sphere. What is the sphere's radius?

Answer

Solving this problem requires recognizing that since the cube is circumscribed by the sphere, both solids share the same center. Now it is just a matter of finding the diagonal of the cube, which will double as the diameter of the sphere (by definition, any straight line which passes through the center of the sphere). The formula for the diagonal of a cube is $D = s\sqrt{3}$ , where is the length of the side of a cube. (This occurs because you must use the Pythagorean theorem once for each 2-dimensional "corner" you travel to find the diagonal for a 3-dimensional shape, but for the ACT it's much faster to memorize the formula.)

In this case:

$D = s\sqrt3 = 25\sqrt3$

Since the radius is half the diameter, divide the result in half:

$r = \frac{D}{2} = \frac{25\sqrt{3}}{2}$

$r = \frac{25\sqrt{3}}{2}$

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Question

Find the surface area of a sphere whose radius is .

Answer

To solve, simply remember the following formula:

$SA=4\pi{r^2}=4\pi3^2=36\pi$

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Question

Find the surface area of a sphere whose radius is . Thus,

Answer

To find surface area, simply use the formula. Thus,

$SA=4\pi{r^2}=4\cdot\pi\cdot3^2=4\cdot\pi\cdot9=36\pi$

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Question

What is the surface area of a composite figure of a cone and a sphere, both with a radius of 5 cm, if the height of the cone is 12 cm? Consider an ice cream cone as an example of the composite figure, where half of the sphere is above the edge of the cone.

Answer

Calculate the slant height height of the cone using the Pythagorean Theorem. The height will be the height of the cone, the base will be the radius, and the hypotenuse will be the slant height.

$s=\sqrt{169}=13$

The surface area of the cone (excluding the base) is given by the formula $A=\pi rs$ . Plug in our values to solve.

$A_{cone}=\pi(5)(13)=65\pi\ cm^2$

The surface area of a sphere is given by $A=4\pi r^2$ but we only need half of the sphere, so the area of a hemisphere is $A=2\pi r^2$ .

$A_{hemisphere}=2\pi(5)^2=50\pi\ cm^2$

So the total surface area of the composite figure is $65\pi\ cm^2+50\pi\ cm^2=115\pi\ cm^2$ .

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Question

Find the surface area of a sphere whose side diameter is .

Answer

To solve, simply use the following formula for the surface area of a circle. Thus,

$SA=4\pi{r^2}=4\pi1^2=4\pi$

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Question

The volume of a sphere is found using the formula $V=\frac{4}{3}\pi r^3$ .

The surface area of a sphere is found using the formula $SA=4\pi r^2$ .

Suppose a sphere has a surface area of $16\pi cm^2$ . What is its volume?

Answer

The first step is to use the surface area formula to find the radius of the sphere.

$A=4\pi r^2$

$16\pi=4\pi r^2$

The next step is to plug the value of the radius into the volume formula.

$V=\frac{4}{3}\pi r^3$

$V=\frac{4}{3}\pi (2)^3$

$V=\frac{4}{3}\pi(8)$

$V=\frac{32}{3}\pi cm^3$

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Question

What is the surface area of a sphere with a diameter, in centimeters, $\small \small \small d=6$ ?

The surface area (SA) of a sphere is calculated using the formula $\small \small SA=4\pi r^{2}$ ?

Answer

If $\small d=6$ , then $\small r=3$ . Plug the radius into the equation for surface area to get

$\small \small \small \small SA=4\pi (3)^{2}=36\pi$ .

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Question

What is the surface area of a sphere that has a diameter of eight inches? Reduce any fractions in your answer and leave your answer in terms of $\pi$ .

Answer

To find the volume of a sphere area of a sphere plug the radius into the following formula given by :

$SA = 4\pi r^2$ .

To find the radius given the diameter, divide the diameter by 2.

$r=\frac{d}{2}$

$r=\frac{8}{2}=4$
Thus:
$SA = 4\pi 4^2 = 64 \pi inches^2$

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Question

What is the surface area of a sphere, in inches, that has a surface area equal to its volume? Leave your answer in terms of $\pi$ .

Answer

To find the radius of a sphere that has a volume equal to it's surface area, begin by setting the volume and surface area formulas for a sphere equal to each other and solving for the radius:
$\\frac{4}{3}\pi r^3 = 4\pi r^2 ewline ewline \frac{4}{3}r^3 = 4r^2 ewline ewline 4r^3 = 12r^2 ewline r^3 = 3r^2 ewline r = 3$

Next we plug the answer for our radius into the formula for the surface area:

$4\pi 3^2 = 36\pi in^2$ , remember to check the units.

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Question

The specifications of an official NBA basketball are that it must be 29.5 inches in circumference and weigh 22 ounces. What is the approximate volume of the basketball? Remember that the volume of a sphere is calculated by V=(4πr3)/3

Answer

To find your answer, we would use the formula: C=2πr. We are given that C = 29.5. Thus we can plug in to get \[29.5\]=2πr and then multiply 2π to get 29.5=(6.28)r. Lastly, we divide both sides by 6.28 to get 4.70=r. Then we would plug into the formula for volume V=(4π〖(4.7)〗3) / 3 (The information given of 22 ounces is useless)

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