A rectangle is inscribed with its base on the x-axis and its upper corners on the parabola y=289−x^2. What are the dimensions of the rectangle with the greatest area?

Question

Give the length of the rectangle as the dimension along the x axis and the width as the vertical height of the rectangle. Enter an exact answer.

Answer 1

Let A=2xy so dA/dx = 2y +2xy' =0. So y'= -2x from y=289 = x² using this in dA/dx =2(289- x²) - 4x² =0 Solving this gives 6x² =578 and this gives x=± 9.81. So 2x= 19.62 and y= 192.67. Taking the second derivative of y' is negative for a positive value of x indicating that the values are for a maximum.