A rectangle is constructed with its base on the x-axis and two of its vertices on the parabola yx. What are the dimensions of the rectangle with the maximum area? What is that area?

The answer is below

Step-by-step explanation:

The question is not complete, the complete question is:

A rectangle is constructed with its base on the x-axis and two of its vertices on the parabola y 81-x², what are the dimensions of the rectangle with the maximum area? what is the area?

Solution:

Given the parabola:

y = 81 - x²

Let the points (a,0) and (-a, 0) be points on the axis, this points touch the parabola at (a, 81 - a²) and (-a, 81 - a²).

Points (a,0), (-a, 0), (a, 81 - a²), (-a, 81 - a²) form the rectangle.

The length of rectangle = distance between (a,0) and (-a, 0) or between (a, 81 - a²) and (-a, 81 - a²) = 2a

The breadth of rectangle = distance between (a,0) and (a, 81 - a²) or between (-a, 0) and (-a, 81 - a²) = 81 - a²

Area of rectangle (A) = length × breadth = 2a × (81 - a²) = 162a - 2a³

A = 162a - 2a³

The maximum area is at dA / da = 0

dA / da = 162 - 8a² = 0

162 - 6a² = 0

6a² = 162

a² = 27

a = 5.2 units

Length = 2a = 2(5.2) =  10.4 units

Breadth = 81 - a² = 81 - 5.2² = 54 unit

Area = length × breadth = 54 × 10.4 = 561 unit²