find the height and width

Question

A rectangle is inscribed with its base on the x-axis and its upper corners on the parabola y=4-x². What are the dimensions of such a rectangle with the greatest possible area?

 

Height=

Width=

Answer 1

We need to label our rectangle with dimension of x and y.

 

Let length = x

Let height = y

 

Area = xy

 

Your equation is      y = 4 - x²

 

 

Lets substitute this value of y into the area equation.

 

Area = x(4 - x²)

Area = 4x - x³

 

Set the derivative of this area equal to zero.

 

4 - 3x² = 0

 

  -3x² = -4

 

     x² = 4/3

 

     x = -1.33        and      x = 1.33

 

 

Now we evaluate y at these value.

 

y = 4 - (-4/3)²

y = 4 - (16/9)

y = (36 - 16) / 9

y = 20/9

y = 2.222

 

 

Then,  length = 2(4/3) = 8/3 = 2 2/3 unit

          width = 2/9 = 2 2/9 units

 

 

Note, the points (-4/3, 20/9) and (4/3, 20/9) are the upper corners of the rectangle.  The bottom corners are (-4/3, 0) and (4/3, 0).