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SAT II Math I : Volume

Study concepts, example questions & explanations for SAT II Math I

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Example Questions

Example Question #711 : Geometry

What is the volume of a regular tetrahedron with an edge length of 6?

Possible Answers:

27.8

25.5

26.6

24.2

32.1

Correct answer:

25.5

Explanation:

The volume of a tetrahedron can be solved for by using the equation:

$V= \frac{a^3}{6\sqrt{2}}$

where is the measurement of the edge of the tetrahedron.

This problem can be quickly solved by substituting 6 in for .

$V=\frac{6^3}{6\sqrt{2}}=\frac{216}{6\sqrt{2}}=\frac{36}{\sqrt{2}}$

${\color{Blue} V=25.5}$

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Example Question #13 : How To Find The Volume Of A Tetrahedron

What is the volume of the tetrahedron shown below?

Screen shot 2015 10 21 at 7.16.10 pm

Possible Answers:

$\frac{256}{3\sqrt2}$

512

$\frac{36}{\sqrt2}$

$\frac{256}{6\sqrt2}$

25.63

Correct answer:

$\frac{256}{3\sqrt2}$

Explanation:

The volume of a tetrahedron is $V=\frac{s^3}{6\sqrt2}$ .

This tetrahedron has a side with a length of 8.

$V=\frac{8^3}{6\sqrt2}$ , which becomes $\frac{512}{6\sqrt2}$ .

You can reduce that answer further, so that it becomes

$\frac{256}{3\sqrt2}$ .

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Example Question #21 : 3 Dimensional Geometry

Find the volume of sphere whose radius is 15.

Possible Answers:

$562.5\pi$

$4500\pi$

$1200\pi$

$36000\pi$

$300\pi$

Correct answer:

$4500\pi$

Explanation:

The volume of a sphere is given by the equation:

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

The problem says that the radius is 15, so plug in 15 for r and simplify.

$V=\frac{4}{3} \pi r^3=\frac{4}{3} \pi 15^3=4500\pi$

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Example Question #22 : 3 Dimensional Geometry

Find the volume of a sphere with the diameter of .

Possible Answers:

$\frac{9}{2}\pi$

$9\pi$

$12\pi$

$27\pi$

$\frac{9}{4}\pi$

Correct answer:

$\frac{9}{2}\pi$

Explanation:

Write the equation for the volume of a sphere.

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

The radius is half the diameter, or $\frac{3}{2}$ .

Substitute the radius into the equation.

$V=\frac{4}{3}\pi( \frac{3}{2})^3$

$V=\frac{4}{3}\pi( \frac{27}{8})$

Reduce the fractions.

The answer is: $\frac{9}{2}\pi$

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Example Question #22 : Volume

Determine the volume of a rectangular prism if the length is $\frac{1}{2}$ , width is $\frac{2}{3}$ , and the height is $\frac{5}{6}$ .

Possible Answers:

$\frac{1}{3}$

$\frac{5}{9}$

$\frac{2}{3}$

$\frac{5}{18}$

$\frac{9}{5}$

Correct answer:

$\frac{5}{18}$

Explanation:

Write the formula for the volume of a rectangular prism.

$A=\textup{ (Length) (Width) (Height)}$

Substitute the dimensions into the formula.

$A = \frac{1}{2}\cdot \frac{2}{3}\cdot \frac{5}{6} = \frac{10}{36}$

Reduce this fraction.

$\frac{5\times 2}{18\times 2}$

The answer is: $\frac{5}{18}$

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Example Question #101 : Geometry

The length of a box is half its height and two-thirds its width. The volume of the box is four cubic meters. Give the length of the box to the nearest centimeter.

Possible Answers:

$144 \textrm{ cm}$

None of these

$229\textrm{ cm}$

$175 \textrm{ cm}$

$110 \textrm{ cm}$

Correct answer:

$110 \textrm{ cm}$

Explanation:

Call , , and the length, width, and height of the crate.

The length of the crate is half its height, so

$L = \frac{1}{2} H$

$2 \cdot L = 2 \cdot \frac{1}{2} H$

2 L = H

The length of the crate is two-thirds its width, so

$L = \frac{2}{3} W$

$\frac{3} {2} \cdot L = \frac{3} {2} \cdot \frac{2}{3} W$

$\frac{3} {2} L = W$

The dimensions of the crate in terms of are , $\frac{3} {2} L$ , and . The volume is their product:

V= LWH

Substitute:

LWH = V

$L \cdot \frac{3} {2} L \cdot 2L = 4$

$3 L ^{3} = 4$

Solve for :

$3 L ^{3} = 4$

$\frac{3 L ^{3} }{3} = \frac{4 }{3}$

$L ^{3} = \frac{4 }{3}$

$L =\sqrt[3]{ \frac{4 }{3}}$

$L \approx 1.101$ meters.

Since one meter comprises 100 centimeters, multiply by 100 to convert to centimeters:

$1.101 \cdot 100 = 110.1$ ,

which rounds to 110 centimeters.

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SAT II Math I : Volume

Study concepts, example questions & explanations for SAT II Math I

All SAT II Math I Resources

Example Questions

Example Question #711 : Geometry

Example Question #13 : How To Find The Volume Of A Tetrahedron

Example Question #21 : 3 Dimensional Geometry

Example Question #22 : 3 Dimensional Geometry

Example Question #22 : Volume

Example Question #101 : Geometry

All SAT II Math I Resources

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