a rancher will use 600 m of fencing to build a corral in the shape of a semicircle on top of a rectangle (Figure 9). Find the dimensions that maximize the area of the corral.

first you need to derive an equation for the perimeter. A semi-circle + 3 sides of a rectangle equals 600m

600 = 1/2(pi)d + d + 2h
600 = 2.57d + 2h

now solve the equation for one of your variables. Let's pick h:

h = (600 - 2.57d) / 2 = 300 - 1.285d

Now that we've established the equation in terms of the perimeter, we need to derive an equation in terms of Area:

A = 1/2(pi)(d/2)^2 + dh

now plug in the equation we found previously for d to eliminate a variable:

A = .39d^2 + d(300 - 1.285d) = .39d^2 + 300d - 1.285d^2 = -.895d^2 + 300d

now we have an Area equation in terms of 1 variable. Now to maximize the area, we find the derivative and set it equal to zero:

A' = -1.79d + 300 = 0
d = 167.6

now that we've found one of the variables, plug this value back into your original Perimeter equation to fing your h value:

600 = 2.57(167.6) + 2h
h = 84.63

So the semi-circle will have a diameter of 167.6 m
and the rectangle will have length of 2 sides being 84.63 m, with the last side being 167.6 m


This problem was done assuming there was not any fence between the semi-circle and one side of the rectangle

thank you, yes the semi circle and the rectangle are attached.