A rectangle is inscribed with its base on the x-axis and its upper corners on the parabola y=27-x^2

Question

A rectangle is inscribed with its base on the x-axis and its upper corners on the parabola     .

a) What are the dimensions of such a rectangle with the greatest possible area?

width = _______________                         length= __________________

b) What is the largest area?

Answer 1

We need a formula for area as a function of the height (y)

parabola: y = 27 - x²

So: the width of the rectangl is simply 2x

and th height is y.

We cn write a formula for the area:

A(x ) = y*2x = (27-x²)* 2x = 54x - 2 x³

To find th maximum area, find the first derivative an set = to 0:

A' = 54 -6x² = 0

x² = 9 and so x = +/-3

finally, A = 54*3 - 2*27 =108

CHECK by plotting using a graphing utility.

Answer 2

Hope you find this video explanation helpful. Feel free to leave me any tips to improve my videos. Thanks!