(a)
The height of PQRS is k units, since it is bounded above by the line y = k.
The length would be |OS| + |OR|, which is exactly the same length as |OQ| + |OP|. At Q, y = k (since it lies on the line y = k), sub this into the formula of the curve and hence x = -sqrt(12 - k) (since q is on left side we take negative). Similarly at P, x = sqrt(12 - k).
So the length of the rectangle is 2sqrt(12 - k)
Hence the area is A = 2ksqrt(12 - k)
(b) Find A' for min/max.
A' = 2sqrt(12 - k) - k/sqrt(12 - k)
A' = 0
2sqrt(12 - k) = k/sqrt(12 - k)
k = 2(12 - k)
k = 24 - 2k
3k = 24
k = 8
If you find the second derivate and plug k = 8 in you will find it is a maximum. Hence it is not a minimum. So the minimum must occur at one of the endpoints. At k = 3, A = 18. At k = 10, A = 28.28.... Hence the minimum value is A = 18 at the point where k = 3
