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High School Math : Solid Geometry

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Example Questions

Example Question #6 : How To Find The Volume Of A Sphere

To the nearest tenth of a cubic centimeter, give the volume of a sphere with surface area 1,000 square centimeters.

Possible Answers:

$3,141.6\textrm{ cm}^{3}$

$2,973.5 \textrm{ cm}^{3}$

$785.4\textrm{ cm}^{3}$

$743.4 \textrm{ cm}^{3}$

$991.2\textrm{ cm}^{3}$

Correct answer:

$2,973.5 \textrm{ cm}^{3}$

Explanation:

The surface area of a sphere in terms of its radius is

$A = 4\pi r^2$

Substitute A = 1,000 and solve for :

$4\pi r^2 =A$

$4\pi r^2 =1,000$

$r^2 =\frac{1,000}{4\pi } = \frac{250}{\pi }$

$r =\sqrt{ \frac{250}{\pi }} \approx 8.92 \textrm{ cm }$

Substitute for in the formula for the volume of a sphere:

$V = \frac{4\pi r^3}{3} \approx\frac{4\pi \cdot 8.92^3}{3} \approx 2,973.5 \textrm{ cm }^{3}$

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Example Question #4 : How To Find The Volume Of A Sphere

Find the volume of the following sphere.

Sphere

Possible Answers:

$270 \pi m^3$

$312 \pi m^3$

$300 \pi m^3$

$288 \pi m^3$

$360 \pi m^3$

Correct answer:

$288 \pi m^3$

Explanation:

The formula for the volume of a sphere is:

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

where is the radius of the sphere.

Plugging in our values, we get:

$V=\frac{4}{3} \pi (6m)^3$

$V=\frac{4}{3} \pi 216m^3 = 288 \pi m^3$

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Example Question #2 : How To Find The Volume Of A Sphere

Find the volume of the following sphere.

Sphere

Possible Answers:

$952 \pi m^3$

$992 \pi m^3$

$982 \pi m^3$

$962 \pi m^3$

$972 \pi m^3$

Correct answer:

$972 \pi m^3$

Explanation:

The formula for the volume of a sphere is:

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

Where is the radius of the sphere

Plugging in our values, we get:

$V = \frac{4 \pi (9m)^3}{3}$

$V = 972 \pi m^3$

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Example Question #1 : How To Find The Volume Of A Sphere

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

Possible Answers:

434.19 cu.in.

138.43 cu.in.

92.48 cu.in.

3468.05 cu.in.

8557.46 cu.in.

Correct answer:

434.19 cu.in.

Explanation:

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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Example Question #1 : Spheres

What is the volume of a sphere with a diameter of ?

Possible Answers:

$288\pi$

$1152\pi$

$2304\pi$

$24\pi$

$48\pi$

Correct answer:

$288\pi$

Explanation:

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

The problem gives us the diameter, however, and not the radius. Since the diameter is twice the radius, or d=2r , we can find the radius.

12=2r

$\frac{12}{2}=r$

6=r .

Now plug that into our initial equation.

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

$V=\frac{4}{3}\pi (6)^3$

$V=\frac{4}{3}\pi(216)$

$V=288\pi$

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Example Question #1 : How To Find The Volume Of A Sphere

The radius of a sphere is . What is the approximate volume of this sphere?

Possible Answers:

$288\pi$

$516\pi$

$20\pi$

$300\pi$

$138\pi$

Correct answer:

$288\pi$

Explanation:

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

$V=\frac{4}{3}\pi(6)^3$

$V=\frac{4}{3}\pi(216)$

$V=288\pi$

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Example Question #1992 : High School Math

A cube has a side dimension of 4. A sphere has a radius of 3. What is the volume of the two combined, if the cube is balanced on top of the sphere?

Possible Answers:

$16 + 36\pi$

$16 + 6\pi$

$4 + 36\pi$

$64 + 36\pi$

$64 + 6\pi$

Correct answer:

$64 + 36\pi$

Explanation:

$V_{cube}=(\text{side})^3=(4)^3=64$

$V_{sphere}=\frac{4}{3}\pi r^3=\frac{4}{3}\pi(3^3)=\frac{4}{3}\pi(27)=36\pi$

$V_{total}=64+36\pi$

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Example Question #1021 : Act Math

What is the volume of a sphere with a diameter of 6 in?

Possible Answers:

$36\pi\ in^{3}$

$72\pi\ in^{3}$

$108\pi\ in^{3}$

$288\pi\ in^{3}$

$216\pi\ in^{3}$

Correct answer:

$36\pi\ in^{3}$

Explanation:

The formula for the volume of a sphere is:

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

where = radius. The diameter is 6 in, so the radius will be 3 in.

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Example Question #1994 : High School Math

A solid hemisphere has a radius of length r. Let S be the number of square units, in terms of r, of the hemisphere's surface area. Let V be the number of cubic units, in terms of r, of the hemisphere's volume. What is the ratio of S to V?

Possible Answers:

$\frac{4r}{3}$

$\frac{9r}{2}$

$\frac{3}{r}$

$\frac{9}{2r}$

$3r$

Correct answer:

$\frac{9}{2r}$

Explanation:

First, let's find the surface area of the hemisphere. Because the hemisphere is basically a full sphere cut in half, we need to find half of the surface area of a full sphere. However, because the hemisphere also has a circular base, we must then add the area of the base.

$S = \frac{1}{2}\cdot$ (surface area of sphere) + (surface area of base)

The surface area of a sphere with radius r is equal to $4\pi r^2$ . The surface area of the base is just equal to the surface area of a circle, which is $\pi r^2$ .

$S=\frac{1}{2}\cdot 4\pi r^2+\pi r^2=2\pi r^2+\pi r^2=3\pi r^2$

The volume of the hemisphere is going to be half of the volume of an entire sphere. The volume for a full sphere is $\frac{4}{3}\pi r^3$ .

$V = \frac{1}{2}\cdot$ (volume of sphere)

$V = \frac{1}{2}\cdot \frac{4}{3}\pi r^3=\frac{2}{3}\pi r^3$

Ultimately, the question asks us to find the ratio of S to V. To do this, we can write S to V as a fraction.

$\frac{S}{V}=\frac{3\pi r^2}{\frac{2}{3}\pi r^3}$

In order to simplify this, let's multiply the numerator and denominator both by 3.

$\frac{S}{V}=\frac{3\pi r^2}{\frac{2}{3}\pi r^3}$ = $\frac{9\pi r^2}{2\pi r^3}=\frac{9}{2r}$

The answer is $\frac{9}{2r}$ .

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Example Question #3 : How To Find The Volume Of A Sphere

If the diameter of a sphere is , find the approximate volume of the sphere?

Possible Answers:

$205\pi$

$167\pi$

$143\pi$

$334\pi$

$95\pi$

Correct answer:

$167\pi$

Explanation:

The volume of a sphere = $\frac{4}{3}\pi r^{3}$

Radius is $\frac{1}{2}$ of the diameter so the radius = 5.

$V=\frac{4}{3}\pi5^{3}$

or $V=\frac{500}{3\pi}$

which is approximately $167\pi$

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All High School Math Resources

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High School Math : Solid Geometry

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #6 : How To Find The Volume Of A Sphere

Example Question #4 : How To Find The Volume Of A Sphere

Example Question #2 : How To Find The Volume Of A Sphere

Example Question #1 : How To Find The Volume Of A Sphere

Example Question #1 : Spheres

Example Question #1 : How To Find The Volume Of A Sphere

Example Question #1992 : High School Math

Example Question #1021 : Act Math

Example Question #1994 : High School Math

Example Question #3 : How To Find The Volume Of A Sphere

All High School Math Resources

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