Question

A rectangle has one of its sides on the x-axis and corners of the opposite side on the piece of the parabola y = 12 − x 2 above the x-axis. (a) (6 points) Let (x, 0) be the coordinates of the lower right vertex of the rectangle. Find an expression A(x) for the area of the rectangle.

Answer 1

`A(x) = 24x - 2x^3`

Explanation:

first we need to visualize what the question is trying to represent

• A rectangle with one of its sides on the x-axis
• The corners of the opposite two sides of the rectangle intersect with a parabola

• the equation of the parabola is
  `y = 12 - x^2`

the equation of the parabola can also be written as

`y = -x^2 + 12` this shows that the parabola has a
`\cap` shape and the turning point (peak) lies at the y-axis at y = 12. The parabola is symmetric along the y-axis!

• right corner of the lower side of the rectangle has the coordinates (x,0)

we can also say that the coordinates of the lower left side are (-x,0). This is because the parabola is symmetric along the y-axis.

Solution:

We have enough information to express the sides lengths of the rectangle:

1) Lower side, (L)

We'll name the lower side 'L' and since this is a rectangle, this side length is the same as the upper side.

The lower side coordinates are (-x,0) and (x,0). Hence the side length is the difference of the two coordinates.

`L = x-(-x)`

`L = 2x`

2) The Left and Right side have the same height (H)

since the the lower side is on the x-axis (y=0), the only side moving is the upper one, i.e the coordinates of the upper corners are (0,y) where
`y = 12- x^2`. The side length of either the left or right sides is
`y`

`H = y`

`H = 12 - x^2`

The Area of the rectangle is:

`Area\,\,= L * H`

`Area\,\,= 2x(12 - x^2)`

`Area\,\,= 24x-2x^3`