Find the area of the largest rectangle that has its lower base on the x-axis and its upper vertices on the parabola whose equation is given by Y=12-x^2.

i graphed the equation but its gives me a parabola, so i don't know what to do after that

To find the area of the largest rectangle that fits under the parabola Y = 12 - x^2 with its lower base on the x-axis, we need to maximize the area of the rectangle.

First, let's understand the problem visually. By graphing the equation Y = 12 - x^2, you correctly obtained a parabolic curve. The curve represents the shape of the region above the x-axis where the parabola is positive.

To find the maximum area, we need to locate the rectangle with the largest possible area under the parabola. The bottom side of the rectangle will be on the x-axis, and the top vertices lie on the parabola. Let's break down the solution step by step:

1. Set up an equation for the area of the rectangle:
Area = length × width

2. Determine the length of the rectangle:
The length of the rectangle corresponds to the x-coordinate difference between the two upper vertices on the parabola. Since the lower base lies on the x-axis, the width of the rectangle is fixed and equal to the difference between the x-coordinates of the upper vertices.

3. Find the coordinates of the upper vertices on the parabola:
Set Y = 0 to find the x-coordinates of the two intersections of the parabola with the x-axis.
12 - x^2 = 0
Solve for x: x = ±√12

4. Calculate the length of the rectangle:
The length is the absolute difference between the two x-coordinates of the upper vertices under the curve:
Length = |(√12) - (-√12)| = 2√12

5. Calculate the width of the rectangle:
The width is the distance between the two y-axis intercepts of the parabola. Set x = 0 and solve for y:
Y = 12 - (0)^2
Width = 12

6. Calculate the area of the rectangle:
Area = Length × Width = 2√12 × 12 = 24√12

The maximum area of the rectangle that fits under the parabola Y = 12 - x^2 is 24√12 square units.