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HSPT Math : Concepts

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Example Questions

Example Question #1331 : Concepts

The height of a cylinder is 3 inches and the radius of the circular end of the cylinder is 3 inches. Give the volume and surface area of the cylinder.

Possible Answers:

V=84.78 in^3, A=123.04in^2

V=74.78 in^3, A=103.04in^2

V=74.78 in^3, A=113.10in^2

V=84.78 in^3, A=103.04in^2

V=84.82 in^3, A=113.10in^2

Correct answer:

V=84.82 in^3, A=113.10in^2

Explanation:

The volume of a cylinder is found by multiplying the area of one end of the cylinder (base) by its height or:

$Volume=\pi r^2h$

where is the radius of the circular end of the cylinder and is the height of the cylinder. So we can write:

$Volume=\pi r^2h=\pi \times 3^2\times 3=84.82in^3$

The surface area of the cylinder is given by:

$A=2\pi r^2+2\pi rh$

where is the surface area of the cylinder, is the radius of the cylinder and is the height of the cylinder. So we can write:

$A=2\pi r^2+2\pi rh$

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

$A=18\pi+18\pi$

$A=36\pi$

A=113.10

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Example Question #1332 : Concepts

The end (base) of a cylinder has an area of $16\pi$ square inches. If the height of the cylinder is half of the radius of the base of the cylinder, give the volume of the cylinder.

Possible Answers:

$8\pi$

$16\pi$

$32\pi$

Correct answer:

$32\pi$

Explanation:

The area of the end (base) of a cylinder is $\pi r^2$ , so we can write:

$\pi r^2=16\pi\Rightarrow r^2=16\Rightarrow r=4\ inches$

The height of the cylinder is half of the radius of the base of the cylinder, that means:

$h=\frac{r}{2}=\frac{4}{2}=2\ inches$

The volume of a cylinder is found by multiplying the area of one end of the cylinder (base) by its height:

$Volume=Area\times h=16\pi\times 2=32\pi$

$Volume=\pi r^2h=\pi\times 4^2\times 2=32\pi$

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

We have two right cylinders. The radius of the base Cylinder 1 is $\sqrt{3}$ times more than that of Cylinder 2, and the height of Cylinder 2 is 4 times more than the height of Cylinder 1. The volume of Cylinder 1 is what fraction of the volume of Cylinder 2?

Possible Answers:

$V_{1}=\frac{2}{3}V_{2}$

$V_{1}=V_{2}$

$V_{1}=\frac{3}{2}V_{2}$

$V_{1}=\frac{3}{4}V_{2}$

$V_{1}=\frac{4}{3}V_{2}$

Correct answer:

$V_{1}=\frac{3}{4}V_{2}$

Explanation:

The volume of a cylinder is:

$V=\pi r^2h$

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

So we can write:

$V_{1}=\pi (r_{1})^2h_{1}$

and

$V_{2}=\pi (r_{2})^2h_{2}$

Now we can summarize the given information:

$r_{1}=\sqrt{3}\cdot r_{2}$

$h_{2}=4\cdot h_{1}\Rightarrow h_{1}=\frac{h_{2}}{4}$

Now substitute them in the $V_{1}$ formula:

$V_{1}=\pi (\sqrt{3}r_{2})^2\times \frac{h_{2}}{4}\Rightarrow V_{1}=\frac{3}{4}\pi (r_{2})^2\times h_{2}\Rightarrow V_{1}=\frac{3}{4}V_{2}$

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Example Question #1333 : Concepts

Two right cylinders have the same height. The radius of the base of the first cylinder is two times more than that of the second cylinder. Compare the volume of the two cylinders.

Possible Answers:

$V_{1}=3\cdot V_{2}$

$V_{1}=2\cdot V_{2}$

$V_{1}=\sqrt{2}\cdot V_{2}$

$V_{1}=V_{2}$

$V_{1}=4\cdot V_{2}$

Correct answer:

$V_{1}=4\cdot V_{2}$

Explanation:

The volume of a cylinder is:

$V=\pi r^2h$

where is the radius of the circular end of the cylinder and is the height of the cylinder. So we can write:

$V_{1}=\pi (r_{1})^2h_{1}$

$V_{2}=\pi (r_{2})^2h_{2}$

We know that

$h_{1}=h_{2}$

and

$r_{1}=2r_{2}$ .

So we can write:

$V_{1}=\pi (r_{1})^2h_{1}=\pi (2r_{2})^2\cdot h_{2}=4\pi (r_{2})^2h_{2}\Rightarrow V_{1}=4V_{2}$

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Example Question #1 : Volume Of A Sphere

A car dealership wants to fill a large spherical advertising ballon with helium. It can afford to buy 1,000 cubic yards of helium to fill this balloon. What is the greatest possible diameter of that balloon (nearest tenth of a yard)?

Possible Answers:

$6.2 \;\textrm{yd}$

$9.3 \;\textrm{yd}$

$15.4 \;\textrm{yd}$

$12.4 \;\textrm{yd}$

$7.7 \;\textrm{yd}$

Correct answer:

$12.4 \;\textrm{yd}$

Explanation:

The volume of a sphere, given its radius, is

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

Set V = 1,000 , solve for , and double that to get the diameter.

$1,000= \frac{4\pi r^{3}}{3}$

$r^{3} = \frac{1,000 \cdot 3 }{4\pi} \approx 238.7$

$r \approx \sqrt[3]{238.7} \approx 6.2$

The diameter is twice this, or 12.4 yards.

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Example Question #1 : Volume Of A Three Dimensional Figure

The diameter of a sphere is . Give the volume of the sphere in terms of .

Possible Answers:

$10\pi t^3$

$24\pi t^3$

$30\pi t^3$

$36\pi t^3$

$72\pi t^3$

Correct answer:

$36\pi t^3$

Explanation:

The diameter of a sphere is so the radius of the sphere would be $6t\div 2=3t$

The volume enclosed by a sphere is given by the formula:

$Volume=\frac{4}{3}\pi r^3=\frac{4}{3}\pi \times (3t)^3=\frac{4}{3}\pi\times 27t^3\Rightarrow Volume=36\pi t^3$

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Example Question #1334 : Concepts

A spherical balloon has a diameter of 10 meters. Give the volume of the balloon.

Possible Answers:

723.3m^3

1000m^3

523.6m^3

1023.3m^3

500m^3

Correct answer:

523.6m^3

Explanation:

The volume enclosed by a sphere is given by the formula:

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

where is the radius of the sphere. The diameter of the balloon is 10 meters so the radius of the sphere would be $10\div 2=5$ meters. Now we can get:

$Volume=\frac{4}{3}\pi r^3=\frac{4}{3}\pi\times 5^3=\frac{4}{3}\times \pi \times 125\approx 523.6 m^3$

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Example Question #3 : Volume Of A Three Dimensional Figure

The volume of a sphere is 1000 cubic inches. What is the diameter of the sphere.

Possible Answers:

$10.4\ inches$

$12.4\ inches$

$10\ inches$

$16.4\ inches$

$12\ inches$

Correct answer:

$12.4\ inches$

Explanation:

The volume of a sphere is:

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

Where is the radius of the sphere. We know the volume and can solve the formula for :

$Volume=\frac{4}{3}\pi r^3=1000\Rightarrow r^3=\frac{3\times1000 }{4\pi}\approx 238.85\Rightarrow r\approx 6.2$ inches

So we can get:

$Diameter=2r=2\times 6.2=12.4inches$

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Example Question #1335 : Concepts

Prism

Give the volume of the above box in cubic centimeters.

Possible Answers:

$28,800 \textrm{ cm}^{3}$

$32,400 \textrm{ cm}^{3}$

$216,000 \textrm{ cm}^{3}$

$324,000 \textrm{ cm}^{3}$

$21,600\textrm{ cm}^{3}$

Correct answer:

$324,000 \textrm{ cm}^{3}$

Explanation:

100 centimeters make one meter, so convert each of the dimensions of the box by multiplying by 100.

$0.9 \times 100 = 90$ centimeters

$0.6 \times 100 = 60$ centimeters

Use the volume formula, substituting L = 90, W = H = 60 :

V = LWH

$V = 90 \cdot 60 \cdot 60 = 324,000$ cubic centimeters

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Example Question #1 : Solid Geometry

A cube made of nickel has sidelength 20 centimeters, Nickel has a density of 8.9 grams per cubic centimeter. What is the mass of this cube?

Possible Answers:

$7.12 \textrm{ kg}$

$3.56 \textrm{ kg}$

$71.2 \textrm{ kg}$

$35.6 \textrm{ kg}$

Correct answer:

$71.2 \textrm{ kg}$

Explanation:

The volume of the cube is $20 ^{3} = 20 \times 20 \times 20 = 8,000$ cubic centimeters. Multiply by the number of grams per cubic centimeter to get $8,000 \times 8.9 = 71,200$ grams, or $71,200 \div 1,000 = 71.2$ kilograms.

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HSPT Math : Concepts

Study concepts, example questions & explanations for HSPT Math

All HSPT Math Resources

Example Questions

Example Question #1331 : Concepts

Example Question #1332 : Concepts

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

Example Question #1333 : Concepts

Example Question #1 : Volume Of A Sphere

Example Question #1 : Volume Of A Three Dimensional Figure

Example Question #1334 : Concepts

Example Question #3 : Volume Of A Three Dimensional Figure

Example Question #1335 : Concepts

Example Question #1 : Solid Geometry

All HSPT Math Resources

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