Question

The area of a rectangle is modeled by the graph where y is the area (in square meters) and x is the width (in meters).

What dimensions result in the maximum area?

A rectangle_ meters by_ meters results in the maximum area.

Answer 1

The dimensions that result in the maximum area are therefore;

Width = 3.5

Height = 3.5

Explanation:

The given parameters are;

The area of the rectangle, y = x × h

Where;

x = The width of the rectangle

h = The height of the rectangle

By examination, we have;

x + h = 7

Therefore, x = 7 - h

Which gives;

y = x × (x - 7) = x² - 7·x

y = x² - 7·x

At x = 0, y = 0

At x = 7, y = 0

However, at x = 1, y = -6

Therefore, the function should be, y = -(x² - 7·x)

The maximum occurs where dy/dx = 0, which gives;

dy/dx = d-(x² - 7·x)/dx = 0

dy/dx = 2·x - 7 = 0

2·x - 7 = 0

x = 7/2 = 3.5

On second derivative, we have;

d²y/dx² = 2, which gives a local minimum

x = 3.5 at maximum height

Which give, the maximum height,
`y_(max)` = -(x² - 7·x) at x = 3.5

∴
`y_(max)` = -(3.5² - 7×3.5) = 12.25

The maximum height = 12.25

The dimensions that result in the maximum area are therefore;

Width, x = 3.5

Height, h = 7 - x = 7 - 3.5 = 3.5

Height, h = 3.5.