SAT II Math II : Surface Area

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

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Example Questions

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:

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

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

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

\(\displaystyle 2,446\textup{ ft}^{2}\)

\(\displaystyle 2,694\textup{ ft}^{2}\)

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:

\(\displaystyle 36\pi\)

\(\displaystyle 72\pi\)

\(\displaystyle 18\pi\)

\(\displaystyle 48\pi\)

\(\displaystyle 27\pi\)

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 #2 : Surface Area

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

Possible Answers:

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

\(\displaystyle 16\pi\)

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

\(\displaystyle 4\pi\)

\(\displaystyle 8\pi\)

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:  \(\displaystyle 16\pi\)

Example Question #1 : Surface Area

Find the surface area of a cube if the area of one of the square faces is \(\displaystyle 16\).

Possible Answers:

\(\displaystyle \textup{The answer cannot be determined.}\)

\(\displaystyle 96\)

\(\displaystyle 76\)

\(\displaystyle 72\)

\(\displaystyle 256\)

Correct answer:

\(\displaystyle 96\)

Explanation:

Write the formula for the surface area of a cube.

\(\displaystyle A = 6s^2\)

The area of a square side is already given.  Substitute that as the \(\displaystyle s^2\) term.

\(\displaystyle A=6(16) = 96\)

The answer is:  \(\displaystyle 96\)

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