PSAT Math Multiple-Choice Question 661: Answer and Explanation

Question: 661

A cubic function would be most appropriate when modeling which of the following mathematical relationships?

• A. A sphere's volume and its radius
• B. A circle's circumference and its diameter
• C. A triangle's area and its height
• D. A cube's edge length and its total surface area

Correct Answer: A

Explanation:

(A) A cubic function has a variable raised to the third degree or, in terms of geometry, has 3 dimensions. Therefore, we need a shape that is 3-dimensional. This rules out choices (B) and (C).

Choice (D) may be tempting because a cube is 3-dimensional. However, surface area is actually only 2 dimensions, so this wouldn't be a cubic function.

Choice (A) is correct because volume of a sphere varies proportionally to the cube of its radius. Volume is always in 3 dimensions, hence, the reason its units are always in cubic units.

Alternatively, you could have written out all of the relationships depicted in the answer choices.

The volume of a sphere is given by the formula , which is a cubic function since the radius variable has degree 3. So choice (A) is correct.

The formula for a circle's circumference is C = 2πr = πd, where d represents diameter. This is a linear relationship between d and C, making choice (B) incorrect.

A triangle's area is . This isn't a cubic function, so choice (C) can't be correct.

A cube's surface area is given by the formula SA = 6x², where x represents the length of each side of the cube. Again, this isn't a cubic function since the degree of x is only 2, so choice (D) is incorrect.