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HSPT Math : Concepts

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Example Questions

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

A cylinder has a height that is three times as long as its radius. If the lateral surface area of the cylinder is 54π square units, then what is its volume in cubic units?

Possible Answers:

27π

81π

243π

9π

54π

Correct answer:

81π

Explanation:

Let us call r the radius and h the height of the cylinder. We are told that the height is three times the radius, which we can represent as h = 3r.

We are also told that the lateral surface area is equal to 54π. The lateral surface area is the surface area that does not include the bases. The formula for the lateral surface area is equal to the circumference of the cylinder times its height, or 2πrh. We set this equal to 54π,

2πrh = 54π

Now we substitute 3r in for h.

2πr(3r) = 54π

6πr² = 54π

Divide by 6π

r² = 9.

Take the square root.

r = 3.

h = 3r = 3(3) = 9.

Now that we have the radius and the height of the cylinder, we can find its volume, which is given by πr²h.

V = πr²h

V = π(3)²(9) = 81π

The answer is 81π.

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

What is the volume of a hollow cylinder whose inner radius is 2 cm and outer radius is 4 cm, with a height of 5 cm?

Possible Answers:

80π cm³

50π cm³

60π cm³

100π cm³

20π cm³

Correct answer:

60π cm³

Explanation:

The volume is found by subtracting the inner cylinder from the outer cylinder as given by V = πr_out² h – πr_in² h. The area of the cylinder using the outer radius is 80π cm³, and resulting hole is given by the volume from the inner radius, 20π cm³. The difference between the two gives the volume of the resulting hollow cylinder, 60π cm³.

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

What is the volume of a right cylinder with a circumference of 25π in and a height of 41.3 in?

Possible Answers:

1032.5π in³

4813.33π in³

25812.5π in³

6453.125π in³

3831.34π in³

Correct answer:

6453.125π in³

Explanation:

The formula for the volume of a right cylinder is: V = A * h, where A is the area of the base, or πr². Therefore, the total formula for the volume of the cylinder is: V = πr²h.

First, we must solve for r by using the formula for a circumference (c = 2πr): 25π = 2πr; r = 12.5.

Based on this, we know that the volume of our cylinder must be: π*12.5²*41.3 = 6453.125π in³

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

An 8-inch cube has a cylinder drilled out of it. The cylinder has a radius of 2.5 inches. To the nearest hundredth, approximately what is the remaining volume of the cube?

Possible Answers:

391.33 in³

203.34 in³

157.08 in³

462 in³

354.92 in³

Correct answer:

354.92 in³

Explanation:

We must calculate our two volumes and subtract them. The volume of the cube is very simple: 8 * 8 * 8, or 512 in³.

The volume of the cylinder is calculated by multiplying the area of its base by its height. The height of the cylinder is 8 inches (the height of the cube through which it is being drilled). Therefore, its volume is πr²h = π * 2.5²* 8 = 50π in³

The volume remaining in the cube after the drilling is: 512 – 50π, or approximately 512 – 157.0795 = 354.9205, or 354.92 in³.

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

An 12-inch cube of wood has a cylinder drilled out of it. The cylinder has a radius of 3.75 inches. If the density of the wood is 4 g/in³, what is the mass of the remaining wood after the cylinder is drilled out?

Possible Answers:

3813.3 g

4791.43 g

3193.33 g

2594.11 g

4921.4 g

Correct answer:

4791.43 g

Explanation:

We must calculate our two volumes and subtract them. Following this, we will multiply by the density.

The volume of the cube is very simple: 12 * 12 * 12, or 1728 in³.

The volume remaining in the cube after the drilling is: 1728 – 168.75π, or approximately 1728 – 530.1433125 = 1197.8566875 in³. Now, multiply this by 4 to get the mass: (approx.) 4791.43 g.

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

A hollow prism has a base 5 in x 6 in and a height of 10 in. A closed, cylindrical can is placed in the prism. The remainder of the prism is then filled with gel around the cylinder. The thickness of the can is negligible. Its diameter is 4 in and its height is half that of the prism. What is the approximate volume of gel needed to fill the prism?

Possible Answers:

187.73 in³

203.44 in³

237.17 in³

249.73 in³

103.33 in³

Correct answer:

237.17 in³

Explanation:

The general form of our problem is:

Gel volume = Prism volume – Can volume

The prism volume is simple: 5 * 6 * 10 = 300 in³

The volume of the can is found by multiplying the area of the circular base by the height of the can. The height is half the prism height, or 10/2 = 5 in. The area of the base is equal to πr². Note that the prompt has given the diameter. Therefore, the radius is 2, not 4. The base's area is: 2²π = 4π. The total volume is therefore: 4π * 5 = 20π in³.

The gel volume is therefore: 300 – 20π or (approx.) 237.17 in³.

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

A hollow prism has a base 12 in x 13 in and a height of 42 in. A closed, cylindrical can is placed in the prism. The remainder of the prism is then filled with gel, surrounding the can. The thickness of the can is negligible. Its diameter is 9 in and its height is one-fourth that of the prism. The can has a mass of 1.5 g per in³, and the gel has a mass of 2.2 g per in³. What is the approximate overall mass of the contents of the prism?

Possible Answers:

139.44 g

973.44 g

11.48 kg

13.95 kg

15.22 kg

Correct answer:

13.95 kg

Explanation:

We must find both the can volume and the gel volume. The formula for the gel volume is:

Gel volume = Prism volume – Can volume

The prism volume is simple: 12 * 13 * 42 = 6552 in³

The volume of the can is found by multiplying the area of the circular base by the height of the can. The height is one-fourth the prism height, or 42/4 = 10.5 in. The area of the base is equal to πr². Note that the prompt has given the diameter. Therefore, the radius is 4.5, not 9. The base's area is: 4.5²π = 20.25π. The total volume is therefore: 20.25π * 10.5 = 212.625π in³.

The gel volume is therefore: 6552 – 212.625π or (approx.) 5884.02 in³.

The approximate volume for the can is: 667.98 in³

From this, we can calculate the approximate mass of the contents:

Gel Mass = Gel Volume * 2.2 = 5884.02 * 2.2 = 12944.844 g

Can Mass = Can Volume * 1.5 = 667.98 * 1.5 = 1001.97 g

The total mass is therefore 12944.844 + 1001.97 = 13946.814 g, or approximately 13.95 kg.

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Example Question #1361 : Concepts

Jessica wishes to fill up a cylinder with water at a rate of gallons per minute. The volume of the cylinder is gallons. The hole at the bottom of the cylinder leaks out 0.5 gallons per minute. If there are gallons in the cylinder when Jessica starts filling it, how long does it take to fill?

Possible Answers:

$8\: \text{min}$

$11\: \text{min}$

$5\: \text{min}$

$6\: \text{min}$

$3\: \text{min}$

Correct answer:

$8\: \text{min}$

Explanation:

Jessica needs to fill up gallons at the effective rate of $1.5\:\text{gal/min}$ . $12\: \text{gal}$ divided by $1.5\: \text{gal/min}$ is equal to $8 \:\text{min}$ . Notice how the units work out.

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Example Question #1362 : Concepts

A vase needs to be filled with water. If the vase is a cylinder that is $\dpi{100} \small 12{}''$ tall with a $\dpi{100} \small 2{}''$ radius, how much water is needed to fill the vase?

Possible Answers:

$\dpi{100} \small 12\pi$

$\dpi{100} \small 48\pi$

$\dpi{100} \small 64\pi$

$\dpi{100} \small 96\pi$

$\dpi{100} \small 24\pi$

Correct answer:

$\dpi{100} \small 48\pi$

Explanation:

Cylinder

$\dpi{100} \small V = \pi r^{2}h$

$\dpi{100} \small V = \pi (2)^{2}\times 12$

$\dpi{100} \small V = 4\times 12\pi$

$\dpi{100} \small V = 48\pi$

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Example Question #21 : Cylinders

A cylinder has a base diameter of 12 in and is 2 in tall. What is the volume?

Possible Answers:

$48\pi in^{3}$

$72\pi in^{3}$

$288\pi in^{3}$

$36\pi in^{3}$

$24\pi in^{3}$

Correct answer:

$72\pi in^{3}$

Explanation:

The volume of a cylinder is $V=\pi r^{2}h$

The diameter is given, so make sure to divide it in half.

The units are inches cubed in this example

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HSPT Math : Concepts

Study concepts, example questions & explanations for HSPT Math

All HSPT Math Resources

Example Questions

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

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

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

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

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

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

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

Example Question #1361 : Concepts

Example Question #1362 : Concepts

Example Question #21 : Cylinders

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