y= sqrt(81-x^2)

What length and width should the rectangle have so that its area is a maximum?

(smaller value)

(larger value)

To find the length and width of the rectangle that will result in its maximum area, we can use calculus to optimize the equation.

First, let's rewrite the equation in terms of the variables for length and width. Let the length of the rectangle be y and the width be x. Then we have the equation:

Area of Rectangle = Length x Width = y * x = sqrt(81 - x^2) * x

To find the dimensions that maximize the area, we need to find the critical points of the function. We can do this by taking the derivative of the area equation with respect to x and setting it equal to zero:

d(Area)/dx = 0

Before we proceed, let's simplify the equation by squaring both sides:

(81 - x^2) * x^2 = Area^2

Now, differentiate both sides of the equation using the product rule and chain rule. The derivative of (81 - x^2) will be -2x, and the derivative of x^2 will be 2x:

(-2x) * x^2 + (81 - x^2) * 2x = 0

Simplifying this equation, we have:

-2x^3 + 162x - 2x^3 = 0

Combining like terms, we get:

-4x^3 + 162x = 0

Now, factor out 2x:

2x (-2x^2 + 81) = 0

Set each factor equal to zero:

2x = 0 => x = 0 (smaller value)
-2x^2 + 81 = 0 => x^2 = 81/2 => x = √(81/2) or x = -√(81/2) (larger value)

Since a negative value for x does not make sense in the context of a rectangle's dimensions, we can disregard the negative solution. So, the larger value for x is x = √(81/2).

Now, we can substitute this value of x back into the original equation to find the corresponding value of y:

y = sqrt(81 - x^2)
= sqrt(81 - (√(81/2))^2)
= sqrt(81 - 81/2)
= sqrt(81/2)

Therefore, the length and width of the rectangle that will result in its maximum area are:

Smaller value: x = 0 (a square)
Larger value: x = √(81/2), y = sqrt(81/2)

Note: The smaller value refers to a square, which is a special type of rectangle where all sides are equal. The larger value refers to a rectangle where the width is equal to the square root of half of the difference of the areas of the unit circle and the square.