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:
`5/4=-6k ==> k=-5/24`
`` (The denominator of the right side was 1/4 so multiply both sides by 1/4.)
So we now have ``
Solve this for t:
Substitute the expression for x:
` t^2=(-5/24 * (-12)/((5z)^2)*y)/z`
Substitute the vale of z: