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HSPT Math : Concepts

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Example Questions

Example Question #1391 : Concepts

A perfect cube has a volume of 8 cubic centimeters. If the height, length and width of the cube were doubled, what would be the volume of the cube?

Possible Answers:

$16\:cm^3$

$64\:cm^3$

$24\:cm^3$

$48\:cm^3$

$128\:cm^3$

Correct answer:

$64\:cm^3$

Explanation:

Volume is calculated by height x width x length: $V=h\cdot w \cdot l$

For a cube, the height, width, and length are all the same value, so the equation can be simplified to V=s^3 , where is the length of one edge of the cube.

We know that for the initial cube, $V=8\:cm^3$ , so we can substitute this into the volume equation and solve for the length of one of the cube's sides:

V=s^3

8=s^3

$\sqrt[3]8=\sqrt[3]{s^3}$

2=s

So, one edge of the initial cube is $2\:cm$ long. When doubled, the cube will have edges that are each $4\:cm$ long. We can solve for the final volume of the cube by substituting $4\:cm$ into the equation for the volume of a cube and solving:

V=s^3

V=4^3

$V=4\cdot 4\cdot 4=64\:cm^3$

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Example Question #1 : Cones

An empty tank in the shape of a right solid circular cone has a radius of r feet and a height of h feet. The tank is filled with water at a rate of w cubic feet per second. Which of the following expressions, in terms of r, h, and w, represents the number of minutes until the tank is completely filled?

Possible Answers:

π(r²)(h)/(180w)

180w/(π(r²)(h))

π(r²)(h)/(60w)

π(r²)(h)/(20w)

20w/(π(r²)(h))

Correct answer:

π(r²)(h)/(180w)

Explanation:

The volume of a cone is given by the formula V = (πr²)/3. In order to determine how many seconds it will take for the tank to fill, we must divide the volume by the rate of flow of the water.

time in seconds = (πr²)/(3w)

In order to convert from seconds to minutes, we must divide the number of seconds by sixty. Dividing by sixty is the same is multiplying by 1/60.

(πr²)/(3w) * (1/60) = π(r²)(h)/(180w)

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

A cone has a base radius of 13 in and a height of 6 in. What is its volume?

Possible Answers:

None of the other answers

1352π in³

338π in³

1014π in³

4394π in³

Correct answer:

338π in³

Explanation:

The basic form for the volume of a cone is:

V = (1/3)πr²h

For this simple problem, we merely need to plug in our values:

V = (1/3)π13²* 6 = 169 * 2π = 338π in³

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Example Question #3 : Cones

A cone has a base circumference of 77π in and a height of 2 ft. What is its approximate volume?

Possible Answers:

8893.5π in³

142,296π in³

11,858π in³

71,148π in³

2964.5π in³

Correct answer:

11,858π in³

Explanation:

There are two things to be careful with here. First, we must solve for the radius of the base. Secondly, note that the height is given in feet, not inches. Notice that all the answers are in cubic inches. Therefore, it will be easiest to convert all of our units to inches.

First, solve for the radius, recalling that C = 2πr, or, for our values 77π = 2πr. Solving for r, we get r = 77/2 or r = 38.5.

The height, in inches, is 24.

The basic form for the volume of a cone is: V = (1 / 3)πr²h

For our values this would be:

V = (1/3)π * 38.5² * 24 = 8 * 1482.25π = 11,858π in³

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

What is the volume of a right cone with a diameter of 6 cm and a height of 5 cm?

Possible Answers:

$120\pi \ cm^{3}$

$180\pi \ cm^{3}$

$60\pi \ cm^{3}$

$45\pi \ cm^{3}$

$15\pi \ cm^{3}$

Correct answer:

$15\pi \ cm^{3}$

Explanation:

The general formula is given by $V = 1/3Bh = 1/3\pi r^{2}h$ , where = radius and = height.

The diameter is 6 cm, so the radius is 3 cm.

$V=\frac{\pi 3^2\times 5}{3}=15\pi$

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Example Question #1 : Cones

There is a large cone with a radius of 4 meters and height of 18 meters. You can fill the cone with water at a rate of 3 cubic meters every 25 seconds. How long will it take you to fill the cone?

Possible Answers:

$41\ \textup{minutes}\ 53\ \textup{seconds}$

$60\ \textup{minutes}$

$600\ \textup{minutes}$

$800\ \textup{minutes}$

$3\ \textup{minutes}\ 20\ \textup{seconds}$

Correct answer:

$41\ \textup{minutes}\ 53\ \textup{seconds}$

Explanation:

First we will calculate the volume of the cone

$V=\frac{1}{3}\pi r^{2}h=\frac{1}{3}\pi\cdot 4^{2}\cdot 18=96\pi\ m^{3}$

Next we will determine the time it will take to fill that volume

$96\pi \ m^{3}\times \frac{25\ s}{3\ m^{3}}=800\pi \ s$

We will then convert that into minutes

$800\pi \ s\times \frac{1\ minute}{60\ seconds}=13.\overline{3}\pi \ minutes\approx 41.9\ minutes\approx 41\ minutes\ 53\ seconds$

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

The height of a cylinder is two times the length of the radius of the circular end of a cylinder. If the volume of the cylinder is $16\pi$ , what is the height of the cylinder?

Possible Answers:

$2\sqrt{2}$

Correct answer:

Explanation:

The volume of a cylinder is:

$Volume=\pi r^2h$

where is the radius of the circular end of the cylinder and is the height of the cylinder.

Since h=2r , we can substitute that into the volume formula. So we can write:

$Volume=\pi r^2h$

$16\pi=\pi r^2\cdot (2r)$

$16\pi =2\pi r^{3}$

$16=2r^{3}$

$8=r^{3}$

r=2

So we get:

$h=2r=2\times 2=4$

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

A cylinder has a diameter of inches and a height of inches. Find the volume, in cubic inches, of this cylinder.

Possible Answers:

98.3 in^3

402.1 in^3

308.9 in^3

100.5 in^3

Correct answer:

100.5 in^3

Explanation:

$\text{Volume of cylinder}=\pi \times \text{radius}^2 \times \text{height}$

Since we are given the diameter, divide that value in half to find the radius.

$\text{radius}=8\div2=4$

Now plug this value into the equation for the volume of a cylinder.

$\text{Volume}=\pi \times 4^4 \times 2$

$\text{Volume}=\pi \times 32=100.5$ in^3

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Example Question #1993 : Hspt Mathematics

Find the volume, in cubic inches, of a cylinder that has a radius of inches and a height of inches.

Possible Answers:

2695.3 in^3

737.5 in^3

4456.1 in^3

3308.1 in^3

Correct answer:

3308.1 in^3

Explanation:

The formula to find the volume of a cylinder is $\pi r^2\times height$ .

Now, plug in the given numbers into this equation.

$\text{Volume}=\pi \times 9^2 \times 13$

$\text{Volume}=\pi \times 1053=3308.1$ in^3

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

A sphere has a diameter of inches. What is the volume of this sphere?

Possible Answers:

458.3 in^3

104.7 in^3

523.6 in^3

632.9 in^3

Correct answer:

523.6 in^3

Explanation:

To find the volume of a sphere, use the following formula:

$\text{Volume}=\frac{4}{3}\pi\times r^3$ , where is the radius of the sphere.

Now, because we are given the diameter of the sphere, divide that value in half to find the radius.

$r=10\div2=5$

Now, plug this value into the volume equation.

$\text{Volume}=\frac{4}{3}\pi\times 5^3$

$\text{Volume}=\frac{4}{3}\pi\times 125$

$\text{Volume}=\frac{500}{3}\pi=523.6$ in^3

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HSPT Math : Concepts

Study concepts, example questions & explanations for HSPT Math

All HSPT Math Resources

Example Questions

Example Question #1391 : Concepts

Example Question #1 : Cones

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

Example Question #3 : Cones

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

Example Question #1 : Cones

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

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

Example Question #1993 : Hspt Mathematics

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

All HSPT Math Resources

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