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

Answer:

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

Step-by-step 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:

$Area\,\,= L \times H$

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

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