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HSPT Math : Concepts

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Example Questions

Example Question #1351 : Concepts

A sphere is perfectly contained within a cube that has a volume of 216 units. What is the volume of the sphere?

Possible Answers:

$12\pi$

$36\pi$

$288\pi$

$9\pi$

$48\pi$

Correct answer:

$36\pi$

Explanation:

To begin, we must determine the dimensions of the cube. This is done by solving the simple equation: V=s^3

We know the volume is 216, allowing us to solve for the length of a side of the cube.

216=s^3

Taking the cube root of both sides, we get s = 6.

The diameter of the sphere will be equal to side of the cube, since the question states that the sphere is perfectly contained. The diameter of the sphere will be 6, and the radius will be 3.

We can plug this into the equation for volume of a sphere:

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

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

We can cancel out the 3 in the denominator.

$V=4\pi (3)^2$

Simplify.

$V=4\pi (9)$

$V=36\pi$

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

The surface area of a sphere is $36\pi\ mm^2$ . Find the volume of the sphere in cubic millimeters.

Possible Answers:

$6\pi\ mm^3$

$3\pi\ mm^3$

$9\pi\ mm^3$

$36\pi\ mm^3$

$4\pi\ mm^3$

Correct answer:

$36\pi\ mm^3$

Explanation:

$SA=4\pi r^2$

$36\pi=4\pi r^2$

36=4r^2

9=r^2

3=r

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

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

$V=\frac{4}{3}(27)\pi$

$V=4(9)\pi$

$V=36\pi$

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Example Question #13 : Spheres

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:

$\frac{3}{r}$

$\frac{9}{2r}$

$\frac{9r}{2}$

$\frac{4r}{3}$

$3r$

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 #21 : Prisms

A rectangular box has a length of 2 meters, a width of 0.5 meters, and a height of 3.2 meters. How many cubes with a volume of one cubic centimeter could fit into this rectangular box?

Possible Answers:

3.2 x 10^-3

3.2 x 10⁶

3.2 x 10²

3.2

3,2 x 10³

Correct answer:

3.2 x 10⁶

Explanation:

In order to figure out how many cubic centimeters can fit into the box, we need to figure out the volume of the box in terms of cubic centimeters. However, the measurements of the box are given in meters. Therefore, we need to convert these measurements to centimeters and then determine the volume of the box.

There are 100 centimeters in one meter. This means that in order to convert from meters to centimeters, we must multiply by 100.

The length of the box is 2 meters, which is equal to 2 x 100, or 200, centimeters.

The width of the box is 0.5(100) = 50 centimeters.

The height of the box is 3.2(100) = 320 centimeters.

Now that all of our measurements are in centimeters, we can calculate the volume of the box in cubic centimeters. Remember that the volume of a rectangular box (or prism) is equal to the product of the length, width, and height.

V = length x width x height

V = (200 cm)(50 cm)(320 cm) = 3,200,000 cm³

To rewrite this in scientific notation, we must move the decimal six places to the left.

V = 3.2 x 10⁶ cm³

The answer is 3.2 x 10⁶.

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Example Question #2 : Prisms

A rectangular prism has a length that is twice as long as its width, and a width that is twice as long as its height. If the surface area of the prism is 252 square units, what is the volume, in cubic units, of the prism?

Possible Answers:

432

1728

216

108

Correct answer:

216

Explanation:

Let l be the length, w be the width, and h be the height of the prism. We are told that the length is twice the width, and that the width is twice the height. We can set up the following two equations:

l = 2w

w = 2h

Next, we are told that the surface area is equal to 252 square units. Using the formula for the surface area of the rectangular prism, we can write the following equation:

surface area = 2lw + 2lh + 2wh = 252

We now have three equations and three unknowns. In order to solve for one of the variables, let's try to write w and l in terms of h. We know that w = 2h. Because l = 2w, we can write l as follows:

l = 2w = 2(2h) = 4h

Now, let's substitute w = 2h and l = 4h into the equation we wrote for surface area.

2(4h)(2h) + 2(4h)(h) + 2(2h)(h) = 252

Simplify each term.

16h² + 8h² + 4h² = 252

Combine h² terms.

28h² = 252

Divide both sides by 28.

h² = 9

Take the square root of both sides.

h = 3.

This means that h = 3. Because w = 2h, the width must be 6. And because l = 2w, the length must be 12.

Because we now know the length, width, and height, we can find the volume of the prism, which is what the question ultimately requires us to find.

volume of a prism = l • w • h

volume = 12(6)(3)

= 216 cubic units

The answer is 216.

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Example Question #1 : Finding Volume Of A Rectangular Prism

The dimensions of Treasure Chest A are 39” x 18”. The dimensions of Treasure Chest B are 16” x 45”. Both are 11” high. Which of the following statements is correct?

Possible Answers:

There is insufficient data to make a comparison between Treasure Chest A and Treasure Chest B.

Treasure Chest A has the same surface area as Treasure Chest B.

Treasure Chest B can hold more treasure.

Treasure Chest A can hold more treasure.

Treasure Chest A and B can hold the same amount of treasure.

Correct answer:

Treasure Chest B can hold more treasure.

Explanation:

The volume of B is 7920 in³. The volume of A is 7722 in³. Treasure Chest B can hold more treasure.

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

Carlos has a pool in the shape of a rectangular prism. He fills the pool with a hose that ejects water at a rate of p gallons per minute. The bottom of the pool is A meters wide and B meters long. If there are k gallons in a cubic meter, then which of the following expressions will be equal to the amount of time it takes, in hours, for the water in the pool to reach a height of c meters?

Possible Answers:

ABck/(60p)

ABc/(pk)

60ABcp/k

60ABc/(pk)

ABcp/(60k)

Correct answer:

ABck/(60p)

Explanation:

Fillpool1

Fillpool2

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

The volume of a cylinder is 36π. If the cylinder’s height is 4, what is the cylinder’s diameter?

Possible Answers:

Correct answer:

Explanation:

Volume of a cylinder? V = πr²h. Rewritten as a diameter equation, this is:

V = π(d/2)²h = πd²h/4

Sub in h and V: 36p = πd²(4)/4 so 36p = πd²

Thus d = 6

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

A cylinder has a height of 5 inches and a radius of 3 inches. Find the lateral area of the cylinder.

Possible Answers:

8π

15π

45π

24π

30π

Correct answer:

30π

Explanation:

LA = 2π(r)(h) = 2π(3)(5) = 30π

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

A cylinder has a volume of 20. If the radius doubles, what is the new volume?

Possible Answers:

100

Correct answer:

Explanation:

The equation for the volume of the cylinder is πr²h. When the radius doubles (r becomes 2r) you get π(2r)²h = 4πr²h. So when the radius doubles, the volume quadruples, giving a new volume of 80.

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HSPT Math : Concepts

Study concepts, example questions & explanations for HSPT Math

All HSPT Math Resources

Example Questions

Example Question #1351 : Concepts

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

Example Question #13 : Spheres

Example Question #21 : Prisms

Example Question #2 : Prisms

Example Question #1 : Finding Volume Of A Rectangular Prism

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

Example Question #1 : 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

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