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

Question

A rectangle has its two lower corners on the x-axis and its two upper corners on the parabola y=11-x²

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

width=?

height=?

Answer 1

This problem can be set up as an optimization problem in calculus

 

The points on the parabolic curve can be written as (x, 11- x²)

so the upper right corner will have the coordinates of (x, 11-x²) for some optimal value of x

 

The width = 2x   and the height = 11- x²     again for some optimal value of x

The objective function is the area (which is to me maximized)

  area = width x height =  A =   2x (11-x²)

 

The calculus approach is to compute the derivative of A with respect to x and set this equal to zero

The analytic  derivative of A is 22 - 6 x² 

setting this equal to zero yields x  = sqrt(11/3) =  1.914    , substituting this result back into the expressions for width and height  gives

 

width = 2 sqrt(11/3)  and height = 22/3

 

The graphing calculator approach is to put the function   2x (11 - x²) into Y1 and plot the curve.

Then use 2nd calc ==> maximum  

The result  is  x =  1.914