How do you find part C?
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Answer:
Step-by-step explanation:
The area function is f(x,y) = x*y the product of the values x and y
You want to maximize f under the constraint g(x) = p, with g(x) = 2x + 2y, the perimeter function. Lagrange Multiplier theorem states that, the point (x,y) where the area takes its maximun value should satisfy this equation:
grad(f) (x,y) = λ grad(g) (x,y), for certain λ.
In order to calculate the gradient of each function, you have to calculate the partial derivates fx, fy, gx, gy. The partial derivates can be obtained by deriving f and g assuming the other variable is a constant. For example, in order to obtain fx you assume that y plays like a constant, then
;
;
;
Using this computations, we obtain
Therefore, we have that 2λ = y, 2λ = x
The first thing that you may realize is that x = y = 2λ. Since 2x + 2y = p, then 2x + 2y = 2x + 2x = 4x = p (because x = y). Therefore x = y = p/4, and 2λ = p/4, then λ = p/8.
We can conclude that:
- x = y = p/4 (the side lenght is p/4 for each side, so the rectangle is a sqaure)
- λ = p/8
- grad(f) = grad(g) = (y,x) = (2λ,2λ) = (p/8,p/8)
- area = x*y = p/4 * p/4 = p²/16
I hope i could helped you!