Arectangle 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.

Step-by-step explanation:

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

the equation of the parabola can also be written as

this shows that the parabola has a shape and the turning point (peak) lies at the y-axis at y = 12. The parabola is symmetric along the y-axis!

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.

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 . The side length of either the left or right sides is

The Area of the rectangle is:

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step-by-step explanation: