Problem Solving

BTGmoderatorDC wrote:In the formula \(w = \frac{p}{\sqrt[t]{v}}\), integers p and t are positive constants. If w = 2 when v = 1 and if w = 1/2 when v = 64, then t =

(A) 1
(B) 2
(C) 3
(D) 4
(E) 16



OA C

Source: Official Guide

We can create the equations:

1/2 = p/(^t√64)

(^t√64)/2 = p

And

2 = p/(^t√1)

2(^t√1) = p

Substituting, we have:

(^t√64)/2 = 2(^t√1)

(^t√64) = 4(^t√1)

(^t√64)/(^t√1) = 4

(^t√64) = 4

Thus, t must be 3.

Alternate Solution:

First, we substitute w = 2 and v = 1 into the given equation to obtain:

2 = p / ^t√1

Since t is positive, we know that ^t√1 = 1 for any t. Thus, we have:

2 = p

Now, we substitute w = ½ and v = 64 and p = 2 into the given equation to obtain:

1/2 = 2 / ^t√64

1 = 4 / ^t√64

^t√64 = 4

Raising both sides to the t power, we have:

64 = 4^t

t = 3

Answer: C