U varies directly as the square root of V when U = 24, V = 9, find the value of V when U = 16.

Answer Details

The relationship described in the problem is a direct variation, where U varies directly as the square root of V. In direct variation, when one variable changes, the other variable changes in a specific way. The formula for this kind of relationship is:

U = k√V

where U is directly proportional to the square root of V with k being the constant of proportionality.

From the problem statement, we know that when U = 24, V = 9, and we need to find the constant of proportionality k.

Let's substitute these values into the equation:

24 = k√9

Solving for k, we find:

24 = k * 3
k = 24 / 3
k = 8

Now we have the constant of proportionality, k = 8. To find V when U = 16, we substitute U = 16 into the equation:

16 = 8√V

Solving for √V, we find:

√V = 16 / 8
√V = 2

To find V, we square both sides of the equation:

V = 2²
V = 4

Therefore, the value of V when U = 16 is 4.