# The variable z varies jointly with x and y and inversely with the square of t. Also, z=5 when x=2, y=-3, and t=1/2. Write a function such that f(y)=t when x=(-)12/(5z)^2 and z=2/5.

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We are given that z varies jointly with x and y and varies inversely with the square of t.

Also, when z=5 we have x=2,y=-3 and t=1/2.

We are asked to write a function f(y)=t when `x=(-12)/(5z)^2,z=2/5` .

First, when a variable varies jointly with 2 or more other variables, this means that it varies directly with those variables. If z varies directly with both x and y we can write z=kxy for some real number k.

Second, if a variable varies inversely with another variable, the first variable varies directly with the reciprocal of the other variable. Since z varies inversely with the square of t we have `z=k/t^2` .

Putting these together we get `z=(kxy)/t^2` for some k.

Now we are given values of x,y, and z so we can compute the value of k:

`5=(k(2)(-3))/(1/2)^2`

`5/4=-6k ==> k=-5/24`

`` (The denominator of the right side was 1/4 so multiply both sides by 1/4.)

So we now have ``

`z=(-5/24 xy)/t^2`

Solve this for t:

`t^2=(-5/24 xy)/z`

Substitute the expression for x:

` t^2=(-5/24 * (-12)/((5z)^2)*y)/z`

`t^2=(y/(10z^2))/z`

Substitute the vale of z:

`t^2=(y/(10(2/5)^2))/(2/5)=(25y)/16`

`t=5/4 sqrt(y)`