rectangular graph

Question

A rectangle has one corner on the graph of y=9-x^3, another at the origin, a third on the positive y-axis, and the fourth on the positive x-axis.

 

Express the area A as a function.

For what value of x is A the largest?

What is the domain of A?

Answer 1

The width of the rectangle would be equal to x; the height to y.

 

Since y = 9 - x³, the area of the rectangle will be A = (width)(height) = x(9 - x³).

 

 

 

Plot y = x(9-x³) to get x = 1.31 for the value of x that maximizes the area.

 

 

 

The width and height have to be positive so x ≥ 0 and 9 - x³ ≥ 0. 

 

So 9 - x³ > 0 or 9 > x³ or x < 9^1/3.

 

 

The domain of the x is 0 ≤ x ≤ 9^1/3.