how do i figure out this word problem?

Question

a rectangle has its base on the x-axis and its two upper corners on the parabola y=12-x² What is the largest possible area of the rectangle?

Answer 1

Hi Marliss, 

 

Take a look at the following link to get an idea of where the rectangle is located: https://www.desmos.com/calculator/uzvm1qz83m Move the slider for "a" to see various flavors of the rectangle. You will see that when "a" has a value of 2 the rectangle appears to have its largest area.

 

What we need is a formula for the area of a general rectangle in terms of an x coordinate. The base of the rectangle can be represented by 2x. Its height can be represented by 12 - x². Since the area of the rectangle is lw, we have in terms of x, A(x) = 2x(12 - x²).  Find A', set it equal to 0 and solve for critical numbers. Convince yourself that those critical numbers generate a maximum area.