Another word problem (psat8/9) which seems really hard to me because how am I supposed to get the answers with one number!

Question

A manufacturer makes two different sizes of spherical ball bearings for use in motors. If the radius of the larger ball bearing is twice the radius of the smaller one, then the volume of the larger ball bearing is how many times the volume of the

smaller one?

Answer choices:

A) 2

B) 3

C) 6

D) 8

Answer 1

V=(4/3)π*r³

Let the small radius= x

Therefore the big radius= 2x

(4/3π) is a constant number therefore is not going to make any difference in our comparison.

So small sphere volume= (4/3π)(x³) by substitution (note that we are treating x as a known digit here)

Large sphere volume= (4/3π)(2x)³= (4/3π)(8x³) It is eight times larger is the answer.

You may verify your answer by using values of , 1,2,3, etc. by substituting for x.

Answer 2

The ratio of volumes of similar figures is as the cube of the linear scale factor.

The linear scale factor is 2 / 1. The ratio of the volumes is 8 / 1.

Answer 3

Karafurushi,

Don’t worry, this problem is not as hard as it looks! The formula for finding volume of a sphere is V = 4/3πr³.

Let’s just plug in some numbers right now:

Say the radius of the smaller ball is 5. This means the radius of the bigger ball if 10.

5³ = 125.

10³ = 1000

1000/125 = 8! And this is your answer. :)

Why did we only look at the r³ part? Because the 4/3π part stays the same no matter what the radius is, so it is a constant. When radius doubles, the r³ part is what changes, so that’s the part we focus on.

Why did we substitute in 5 and 10 instead of easier numbers like 1 and 2? College Board knows that students will plug in numbers to figure out these types of problems, so they sometimes make it so that plugging in the obvious choices of 1 or 2 will lead you to the wrong answer. Generally I advise students to plug in 3, but I was just personally feeling like 5 today. Normally, you should plug in 3 if it seems like you have to plug in numbers!

Btw, for this particular problem plugging in 1 and 2 for your initial radius and larger radius would have worked just fine, but I don’t want you to get in the habit of plugging in those numbers during your SAT or PSAT.

Hope this helps! :)