Question

Suppose that y varies directly as x and inversely as the square of z. When x = 3 and z = 5, y = 4.5. Find y when x = –2 and z = 3.
A – 25/3
B 25/3
C –3/25
D 3/25

Answer 1

y varies directly as x and inversely as the square of z can be written as:
\[y=k\frac{x}{z^2}\]
where k is a constant.

Answer 2

"y varies directly as x" means that y is some constant k multiplied by x. "y varies inversely as the square of z" means that y is some constant divided by the square of z. Hence the equation I wrote above.

Answer 3

We have to find the value of the constant k. We know that y=4.5 then x=3 and z=5. Let's plug in these values:
\[y=k\frac{x}{z^2}\]
\[4.5=k\frac{3}{5^2}\]
\[4.5=k\frac{3}{25}\]
\[k=\frac{4.5\times25}{3}=1.5\times25\]
k=37.5

Answer 4

Rewriting the equation with the value we discovered for k:
\[y=37.5\frac{x}{z^2}\]
Now we can find y when x=-2 and z=3:
\[y=37.5\frac{-2}{3^2}\]
\[y=37.5\frac{-2}{9}\]
\[y=-8.3333...\]
The fraction 8.333... is equivalent to 25/3. Therefore:
\[y=-\frac{25}{3}\]
So, the correct answer is A.