tunnel - differentiate

[w33bob]

Ok folks any help with this would be greatly appreciated!

Question:
A tunnel cross-section is to be in the shape of a rectangle, surmounted by a semicircular roof. The total cross-sectional area is to be A, but the perimter minimised to save building costs. Find the dimensions of the tunnel.

Thanks in advance
bob

[anna69]

I think you need to

a) establish A in terms of x & y (width & height of rect; x/2 is radius of top part.

b) ditto establish perimeter P in terms of x & y

c) rearrange a) for y in terms of x & A

d) substitute c) into b) so Perim is in terms of x & A

e) differentiate d) i.e dP/dx

f) set e)=0, solve for x

g) then substitute x back into a) to get y.

More or less off the top of my head.... something like that anyway1