Question

A rectangle has its base on the x-axis and its upper two vertices on the parabola y=12-x^2. What is the largest area the rectangle can have?

Answer 1

since this is symmetrical around the origin, we can just look at the positive part, maximize it, then double it to get the maximum area. So the area will be equal to the base times the height, where the base is the value from 0 to x, or just x, and the height is determined by the parabola. Multiply the two together to get, 12x-x^3. take the derivative set it equal to 0, and that is your value of x.

Answer 2

it should be close to a square

Answer 3

actually, no, it won't resemble a square at all. it is a rectangle with base 4, and height 8

Answer 4

from my first post, you will get x = 2, then multiply that by the value you get for the parabola by plugging in x, and then multiply by 2 to account for the negative half. 32 is your maximum value.