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

**Solution:**

Given:

Equation of Parabola is y = 12 - x^{2} is an even function.

Therefore, its rectangle form also is even at the origin.

We know area of rectangle = length × width

Here,

length = 2x, width = y

Area, A = 2x(12 - x^{2})

⇒ A = 24x - 2x^{3}

Take derivative of A with respect to x

⇒ A' = 24 - 6x^{2}

The area is largest when A' = 0

⇒ 24 - 6x^{2} = 0

⇒ x^{2} = 4

⇒ x = 2

Put the value of x in y = 12 - x^{2}

⇒ y = 12 - 4

⇒ y = 8

Area = 2(2)(8) = 32

Therefore, the largest area of a rectangle is 32 square units.

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

**Summary:**

For a rectangle with its base on the x-axis and its upper two vertices on the parabola y= 12 - x^{2}, the largest area of the rectangle is 32 square units.

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