cross-section, radius and height

[red and white kop!]

the cross section of an object has the shape of a quarter-circle of radius r adjoining a rectangle of width x and height r, as shown in the diagram.
a. the perimeter and area of the cross-section are P and A respectively. express each of P and A in terms of r and x and hence show that A=0.5Pr -r^2
this part was easy, P= 2r+2x+0.5(pi)(r), A= rx + 0.25(pi)(r)^2
b. taking the perimeter P as fixed, find x in terms of r for the case when the area A of the cross section is a maximum, and show that, for this value of x, A is a maximum and not a minimum.

here i don't know where to start, i dont understand how to combine the equations i have. it seems almost impossible to get one differentiable equation with two given equations and 4 variables! can someone work me through this?

 

[Deleted member 4993]

the cross section of an object has the shape of a quarter-circle of radius r adjoining a rectangle of width x and height r, as shown in the diagram.
a. the perimeter and area of the cross-section are P and A respectively. express each of P and A in terms of r and x and hence show that A=0.5Pr -r^2
this part was easy, P= 2r+2x+0.5(pi)(r), A= rx + 0.25(pi)(r)^2
b. taking the perimeter P as fixed, find x in terms of r for the case when the area A of the cross section is a maximum, and show that, for this value of x, A is a maximum and not a minimum.

here i don't know where to start, i dont understand how to combine the equations i have. it seems almost impossible to get one differentiable equation with two given equations and 4 variables! can someone work me through this?

start with setting (dA/dr) = 0 ? 0.5P - 2r = 0 ? P = 4r (this is max. case since A" = -2)

2x = P - 2r - ?/2 * r ? 2x = 4r - 2r - ?/2 * r ? and continue.....