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High School Math : Solid Geometry

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Example Questions

Example Question #21 : Cylinders

Find the volume of the following partial cylinder.

Partial_cylinder

Possible Answers:

$520 \pi m^3$

$530 \pi m^3$

$540 \pi m^3$

$550 \pi m^3$

$560 \pi m^3$

Correct answer:

$540 \pi m^3$

Explanation:

The formula for the volume of a partial cylinder is:

V = (base)(height)(part)

$V = (\pi r^2)(h)\left(\frac{part}{360}\right)$

where is the radius of the cylinder, is the height of the cylinder, and part is the degrees of the sector.

Plugging in our values, we get:

$V=(\pi (6m)^2)(20m)\left(\frac{270}{360}\right)$

$V=540 \pi m^3$

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

If a cylinder has a radius, $\small r$ , of 2 inches and a height, $\small h$ , of 5 inches, what is the total surface area of the cylinder?

Possible Answers:

$\small 18\pi$

$\small 28\pi$

$\small 70\pi$

$\small 36\pi$

$\small 24\pi$

Correct answer:

$\small 28\pi$

Explanation:

The total surface area will be equal to the area of the two bases added to the area of the outer surface of the cylinder. If "unwrapped" the area of the outer surface is simply a rectangle with the height of the cylinder and a base equal to the circumference of the cylinder base. We can use these relationships to find a formula for the total area of the cylinder.

$A=2A_{base}+A_{rectangle}$

$A=2(\pi r^2)+(2\pi r)(h)$

Use the given radius and height to solve for the final area.

$\small 2\pi(2)^{2} + 2\pi (2)(5)$

$\small 8\pi + 20\pi$

$\small 28\pi$

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Example Question #2 : How To Find The Surface Area Of A Cylinder

The volume of a cylinder is $81\pi$ . If the radius of the cylinder is , what is the surface area of the cylinder?

Possible Answers:

$3\pi$

$9\pi$

$81\pi$

$72\pi$

$27\pi$

Correct answer:

$72\pi$

Explanation:

The volume of a cylinder is equal to:

$V=\pi r^2h$

Use this formula and the given radius to solve for the height.

$81\pi=\pi(3)^2h$

$81\pi=9\pi(h)$

h=9

Now that we know the height, we can solve for the surface area. The surface area of a cylinder is equal to the area of the two bases plus the area of the outer surface. The outer surface can be "unwrapped" to form a rectangle with a height equal to the cylinder height and a base equal to the circumference of the cylinder base. Add the areas of the two bases and this rectangle to find the total area.

$A=2A_{base}+A_{rec}$

$A=2\pi r^2+(2\pi r)(h)$

Use the radius and height to solve.

$A=2\pi (3)^2+2\pi (3)(9)$

$A=18\pi+54\pi$

$A=72\pi$

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Example Question #2 : How To Find The Surface Area Of A Cylinder

What is the surface area of a cylinder with a base diameter of 6in and a height of 8in ?

Possible Answers:

$48\pi in^{2}$

$14\pi in^{2}$

$66\pi in^{2}$

$36\pi in^{2}$

None of the answers

Correct answer:

$66\pi in^{2}$

Explanation:

Area of a circle $A=\pi r^{2}=\pi(\frac{d}{2})^{2}=\pi(\frac{6}{2})^{2}=\pi(3)^{2}=9\pi in^{2}$

Circumference of a circle $C=\pi d=6\pi in^{2}$

Surface area of a cylinder $SA=2A+hC=2(9\pi)+8(6\pi)=18\pi+48\pi=66\pi in^{2}$

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Example Question #26 : Cylinders

A balloon, in the shape of a sphere, is filled completely with water. The surface area of the filled balloon is $144\pi$ . If all of the balloon's water was emptied into a cylindrical cup of base radius 4, how high would the water level be?

Possible Answers:

$4\pi$

Correct answer:

Explanation:

The surface area of a sphere is given by the formula

$S (sphere) = 4\pi*r^2$ , which here equals $144\pi$ . So

$4\pi*r^2 = 144\pi$

r^2 = 36

r = 6

To find out what the water level would be, we need to know how much water is in the balloon. So we need to find the volume of the balloon.

The amount of water that is in the balloon is

$V(sphere) = \frac{4\pi*r^3}{3}$

= $\frac{4\pi*6^3}{3}$

$288\pi$

The cylindrical column of water will have this volume, and will have base radius 4. Using the formula for volume of a cylinder, we can solve for the height of the column of water.

$V(cylinder) = \pi*r^2*h$ , where is the base radius 4 in this case.

$288\pi = \pi*(4^2)*h$

18 = h

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

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

Possible Answers:

$2304\pi$

$1728\pi$

$144\pi$

$1600\pi$

Correct answer:

$2304\pi$

Explanation:

To solve for the volume of a sphere, you must first know the equation for the volume of a sphere.

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

In this equation, is equal to the radius. We can plug the given radius from the question into the equation for .

$V=\frac{4}{3}(\pi)(12^{3})$

Now we simply solve for .

$V=\frac{4}{3}(\pi)(1728)$

$V=(\pi)(2304)=2304\pi$

The volume of the sphere is $2304\pi$ .

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

What is the volume of a sphere with a radius of 4? (Round to the nearest tenth)

Possible Answers:

$75.9\pi$

$91.2\pi$

$75.3\pi$

$85.3\pi$

Correct answer:

$85.3\pi$

Explanation:

To solve for the volume of a sphere you must first know the equation for the volume of a sphere.

The equation is

Then plug the radius into the equation for yielding

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

Then cube the radius to get

$V=\frac{4}{3}(64)\pi$

Multiply the answer by $\frac{4}{3}$ and to yield $85.3\pi$ .

The answer is $85.3\pi$ .

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

For a sphere the volume is given by V = (4/3)πr³ and the surface area is given by A = 4πr². If the sphere has a surface area of 256π, what is the volume?

Possible Answers:

300π

615π

683π

750π

Correct answer:

683π

Explanation:

Given the surface area, we can solve for the radius and then solve for the volume.

4πr² = 256π

4r² = 256

r² = 64

r = 8

Now solve the volume equation, substituting for r:

V = (4/3)π(8)³

V = (4/3)π*512

V = (2048/3)π

V = 683π

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

Circle_with_diameter

A typical baseball is 76mm in diameter. Find the baseball's volume in cubic centimeters.

Possible Answers:

$22984.7cm^{3}$

$1838cm^{3}$

$230cm^{3}$

$2786cm^{3}$

Not enough information to solve

Correct answer:

$230cm^{3}$

Explanation:

In order to find the volume of a sphere, use the formula

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

We were given the baseball's diameter, $\dpi{100} D=76mm$ , which must be converted to its radius.

D=2r

76mm=2r

$r=\frac{76mm}{2}$

$\rightarrow 38mm$

Now we can solve for volume.

$V=\frac{4}{3}\pi (38mm)^{3}$

$V=\frac{4}{3}\pi (54872mm^{3})$

$V=73162\frac{2}{3}mm^{3}*\pi$

$\rightarrow 229847.30mm^{3}$

Convert to centimeters.

$\dpi{100} \frac{229847.30mm^{3}}{1}*\frac{1cm^{3}}{1000mm^{3}}\approx 230cm^{3}$

If you arrived at $1838cm^{3}$ then you did not convert the diameter to a radius.

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Example Question #5 : How To Find The Volume Of A Sphere

What is the volume of a sphere whose radius is r=1.6in .

Possible Answers:

$\dpi{100} 16.17in^{3}$

$17.16in^{3}$

Not enough information to solve

$\dpi{100} 9.65in^{3}$

$\dpi{100} \dpi{100} 6.59in^{3}$

Correct answer:

$17.16in^{3}$

Explanation:

In order to find the volume of a sphere, use the formula

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

We were given the radius of the sphere, r=1.6in .Therefore, we can solve for volume.

$\dpi{100} V=\frac{4}{3}\pi (1.6in)^{3}$

$\dpi{100} V=\frac{4}{3}\pi (4.096in^{3})$

$\dpi{100} V=17.16in^{3}$

If you calculated the volume to be $\dpi{100} 9.65in^{3}$ then you multiplied by $\frac{3}{4}$ rather than by $\frac{4}{3}$ .

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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 #21 : Cylinders

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

Example Question #2 : How To Find The Surface Area Of A Cylinder

Example Question #2 : How To Find The Surface Area Of A Cylinder

Example Question #26 : Cylinders

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

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

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

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

Example Question #5 : How To Find The Volume Of A Sphere

All High School Math Resources

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