Expression for h is h = 500/(pi * r) - r
Range of values for r = (0, 10sqrt(5/pi)]
First, let's substitute the maximum area of the greenhouse into the provided equation.
500 = pi * r * h + pi * r^2
Now solve for h
500 = pi * r * h + pi * r^2
500 - pi * r^2 = pi * r * h
(500 - pi * r^2) / pi * r = h
500/(pi * r) - r = h
The minimum value for r is just above 0, since at 0, you're attempting to divide by 0.
The maximum value for r is where h = 0, so let's substitute 0 for h and solve for r, giving
500/(pi * r) - r = h
500/(pi * r) - r = 0
500/(pi * r) = r
500 = pi * r^2
500/pi = r^2
sqrt(500/pi) = r
10sqrt(5/pi) = r
So maximum r is approximately 12.616 ft.
The range of values for r is (0, 10sqrt(5/pi)]
e.g. You're not allowed to quite reach the lower limit since that would attempt to divide by 0, but you're allowed to go all the way to the upper limit.
