You want to build a greenhouse from at most 500 square feet of material in the shape of a half cylinder. the surface area of the greenhouse is given by the formula s = \pi rh + \pi r^2 where r is the radius and h is the length of the greenhouse. write an expression for h in the terms of r. then find all of the possible values of r

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.