Find the dimensions of the largest rectangle that can be inscribed in the parabolic curve below and the X-axis : y=12 - x^2

Question

Calculus help

Answer 1

Let length of rectangle = x

Let width of rectangle = y

 

Area = xy

 

Substitute the parabolic equation into the area.

 

Area = x(12 - x²)

Area = 12x - x³

 

Set the derivative of area equal to zero.

 

12 - 3x² = 0

 

3(4 - x²) = 0

 

x = -2           and            x = 2

 

The largest length is bounded by x=-2 and x=2.  Therefore, the length is 4 units.

 

Next, we find the width of the rectangle.

 

width = 12 - 2²

width = 8 units