A rectangle is inscribed with its base on the x-axis and its upper corners on the parabola y = 5 - x^2 . What are the dimensions of such a rectangle with the greatest possible area?

Question

Applied Optimization

A rectangle is inscribed with its base on the x-axis and its upper corners on the parabola y = 5 - x². What are the dimensions of such a rectangle with the greatest possible area?

Width = ____

Height = ____

And please explain, I have an exam tomorrow, and applied optimization will be on it and I don't understand it.

Answer 1

Let say one corner is (x, 0 and next corner is (x, 5 - x²). Then area A = x(5 - x²).

Then area A = x(5 - x²) = 5x - x³; A' = 5 - 3x² = 0; x = ±√5/3; A'' = - 6x; at x = √5/3 A'' < 0. So, we have max

width = 2x = 2√5/3 = 2.582

height = y = √5/3(5 - 5/3) = 10(√5/3)/3 = 4.3