SAT II Math II : SAT Subject Test in Math II

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

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Example Questions

Example Question #401 : Sat Subject Test In Math Ii

Determine the volume of a sphere with a diameter of 6.

Possible Answers:

\displaystyle 296\pi

\displaystyle 36\pi

\displaystyle 24\pi

\displaystyle 396\pi

\displaystyle 48\pi

Correct answer:

\displaystyle 36\pi

Explanation:

The radius is half the diameter, which is three.

Write the formula for the volume of a sphere.

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

Substitute the radius.

\displaystyle V=\frac{4}{3}\pi (3)^3 = 36\pi

The answer is:  \displaystyle 36\pi

Example Question #12 : Volume

Find the volume of a sphere with a radius of \displaystyle 2\pi^4.

Possible Answers:

\displaystyle \frac{16}{3}\pi ^{12}

\displaystyle \frac{16}{3}\pi ^{13}

\displaystyle \frac{4}{3}\pi ^{13}

\displaystyle \frac{32}{3}\pi ^{13}

\displaystyle 8\pi ^{13}

Correct answer:

\displaystyle \frac{32}{3}\pi ^{13}

Explanation:

Write the formula for the sphere.

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

Substitute the radius.

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

The answer is:  \displaystyle \frac{32}{3}\pi ^{13}

Example Question #21 : Volume

Find the volume of a sphere with a surface area of \displaystyle 36\pi.

Possible Answers:

\displaystyle 18\pi

\displaystyle 72\pi

\displaystyle 27\pi

\displaystyle 36\pi

\displaystyle 96\pi

Correct answer:

\displaystyle 36\pi

Explanation:

Write the surface area formula for a sphere.

\displaystyle A= 4\pi r^2

Substitute the area.

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

Divide both sides by \displaystyle 4\pi.

\displaystyle \frac{36\pi}{4\pi}=\frac{ 4\pi r^2}{4\pi}

\displaystyle r^2 =9

\displaystyle r=3

Write the formula for the volume of a sphere.

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

Substitute the radius.

\displaystyle V=\frac{4}{3}\pi (3)^3 = 36\pi

The answer is:  \displaystyle 36\pi

Example Question #1 : Surface Area

A rectangular swimming pool is \displaystyle D meters deep throughout and \displaystyle W meters wide. Its length is ten meters greater than twice its width. Which of the following expressions gives the total surface area, in square meters, of the inside of the pool?

Possible Answers:

\displaystyle 4W^{2} + 3DW + 10W+ 20D

\displaystyle 4W^{2} + 6DW + 20W+ 20D

\displaystyle 2W^{2} + 6DW + 10W+ 20D

\displaystyle 2DW^{2}+ 10DW

\displaystyle 2W^{2} + 3DW + 10W+ 10D

Correct answer:

\displaystyle 2W^{2} + 6DW + 10W+ 20D

Explanation:

Since the length of the pool is ten meters longer than twice its width \displaystyle W, its length is \displaystyle 2W + 10.

The inside of the pool can be seen as a rectangular prism. The bottom, or the base, has dimensions \displaystyle W and \displaystyle 2W + 10, so its area is the product of these:

\displaystyle W (2W + 10) = 2W^{2}+ 10W

The sides of the pool have depth \displaystyle D. Two sides have width \displaystyle W and therefore have area \displaystyle DW.

Two sides have width \displaystyle 2W + 10 and therefore have area 

\displaystyle D \left (2W + 10 \right ) = 2DW + 10D

The total area of the inside of the pool is

\displaystyle 2W^{2}+ 10W + 2 (DW ) + 2 (2DW + 10D)

\displaystyle =2W^{2}+ 10W + 2 DW + 4DW + 20D

\displaystyle =2W^{2} + 6DW + 10W+ 20D

Example Question #2 : Surface Area

A water tank takes the shape of a closed rectangular prism whose exterior has height 30 feet, length 20 feet, and width 15 feet. Its walls are one foot thick throughout. What is the total surface area of the interior of the tank?

Possible Answers:

\displaystyle 2,204 \textrm{ ft}^{2}

\displaystyle 1,223 \textrm{ ft}^{2}

\displaystyle 1,102\textrm{ ft}^{2}

Correct answer:

\displaystyle 2,204 \textrm{ ft}^{2}

Explanation:

The height,  length, and width of the interior tank are each two feet less than the corresponding dimension of the exterior of the tank, so the dimensions of the interior are 28, 18, and 13 feet. The surface area of the interior is what we are looking for here. It comprises six rectangles:

Two with area \displaystyle 28 \times 18 = 504 square feet;

Two with area \displaystyle 28 \times 13 = 364 square feet;

Two with area \displaystyle 18 \times 13 = 234 square feet.

Add:

\displaystyle 2 \times 504 + 2 \times 364 + 2 \times 234 = 1,008 + 728 + 468 = 2,204 square feet.

Example Question #1 : Surface Area

What is the surface area of a cube with a side length of 5?

Possible Answers:

\displaystyle 30

\displaystyle 75

\displaystyle 150

\displaystyle 25

\displaystyle 125

Correct answer:

\displaystyle 150

Explanation:

If you were to take apart a cube so that you could lay it flat on a surface, you would be able to see that a cube is just made up of 6 identical squares. The area of one square is the length of the side squared, so the surface area of the cube would be denoted with the formula:

\displaystyle A=6*s^2

In this case the side length is 5, so plugging that into the formula will get the answer.

\displaystyle A=6*(5)^2=6*25=150

Example Question #1 : Surface Area

Find the surface area of a rectangular prism with length, width, and height dimensions of \displaystyle \frac{1}{2}\displaystyle \frac{3}{4}, and \displaystyle \frac{5}{2}, respectively.

Possible Answers:

\displaystyle 7

\displaystyle \frac{15}{16}

\displaystyle 5

\displaystyle \frac{15}{8}

\displaystyle 12

Correct answer:

\displaystyle 7

Explanation:

Be careful not to confuse surface area with volume!

There are 6 faces in a rectangular prism, and we will need the sum of all the areas of each face.

Write the formula.

\displaystyle A=2(LW+LH+WH)

Substitute the dimensions.

\displaystyle A= 2[(\frac{1}{2})(\frac{3}{4})+(\frac{1}{2})(\frac{5}{2})+(\frac{3}{4})(\frac{5}{2})]

Evaluate each product in the bracket.

\displaystyle A= 2[\frac{3}{8}+\frac{5}{4}+\frac{15}{8}] = \frac{3}{4}+\frac{10}{4}+\frac{15}{4}

Combine the terms of the numerator.

\displaystyle A = \frac{28}{4} =7

The answer is:  \displaystyle 7

Example Question #1 : Surface Area

What is the surface area of a cube with a side length of \displaystyle 3x?

Possible Answers:

\displaystyle 27x^3

\displaystyle 54x^2

\displaystyle 27x^2

\displaystyle 9x

\displaystyle 54x^6

Correct answer:

\displaystyle 54x^2

Explanation:

Write the formula for the surface area of a cube.

\displaystyle A= 6s^2

Substitute the side length.

\displaystyle A= 6(3x)^2 = 6(3x)(3x) = 54x^2

The answer is:  \displaystyle 54x^2

Example Question #1 : Surface Area

Find the surface area of a sphere with a radius of 3.

Possible Answers:

\displaystyle 72\pi

\displaystyle 36\pi

\displaystyle 48\pi

\displaystyle 27\pi

\displaystyle 18\pi

Correct answer:

\displaystyle 36\pi

Explanation:

Write the formula for the surface area of a sphere.

\displaystyle A=4\pi r^2

Substitute the radius into the equation.

\displaystyle A=4\pi(3)^2 = 4\pi (9) = 36\pi

The answer is:  \displaystyle 36\pi

Example Question #401 : Sat Subject Test In Math Ii

Find the surface area of a sphere with a radius of 2.

Possible Answers:

\displaystyle 8\pi

\displaystyle 16\pi

\displaystyle \frac{16}{3}\pi

\displaystyle \frac{8}{3}\pi

\displaystyle 4\pi

Correct answer:

\displaystyle 16\pi

Explanation:

Write the formula for the surface area of a sphere.

\displaystyle A=4\pi r^2

Substitute the radius.

\displaystyle A=4\pi (2)^2 = 16\pi

The answer is:  \displaystyle 16\pi

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