A graphing calculator is recommended Find the dimensions tha

A graphing calculator is recommended. Find the dimensions that give the largest area for the rectangle shown in the figure Its base is on the x-axis and its other two vertices ere above the x-axis, tying on the parabola y = k - x^2, k = 8. (Round your answers two decimal places.) height width

Solution

y = k -x^2 = 8 - x^2

y =8 - x^2

Area , A = 2x(8 - x^2)

= 16x - 2x^3

maximum area : find dA/dx = 16 - 6x^2

dA/dx =0 ; 16 - 6x^2 =0 ; x = sqrt(8/3) , -sqrt(8/3) ( critical points)

Check A(x) on these points : find 2nd derivative : f\"(x) = -12x

at x= sqrt8/sqrt3 , f\'(x) = -12(sqrt8/sqrt3) < 0 ( Maximum)

Maximum occurs at x= sqrt8/sqrt3

Width = 2sqrt(8/3) = 3.27 ft

; height , y = 8 -(sqrt8/sqrt3)^2 = 8 - 8/3 = 16/3

Height = 16/3 = 5.33 ft