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A rectangle has its base on the x−axis and its upper vertices on the parabola y = 9−x^2. find the maximum possible area of the rectangle.

User JercSi
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The maximum possible area of the rectangle is 12√3 square units.

How to find the maximum possible area ?

The area "A" of the rectangle is given by:

A = 2x * y

Find an expression for "y" in terms of "x" using the equation of the parabola:

y = 9 - x ²

To find the maximum area, maximize the area function A(x) = 2x * (9 - x ²) by taking its derivative and setting it equal to zero:

A'(x) = 2(9 - x ²) - 2x(2x) = 18 - 2x ² - 4x ² = 18 - 6x ²

Setting A'(x) equal to zero and solving for x:

18 - 6x ² = 0

6x ² = 18

x ² = 3

x = ±√3

Plug this value of x back into the equation for y:

y = 9 - 3

y = 6

So, the dimensions of the rectangle that maximize its area are a base of 2√3 and a height of 6. Now, calculate the maximum area:

A = 2x * y = 2(√3) * 6

= 12√3 square units

Therefore, the maximum possible area of the rectangle is 12√3 square units.

User Mathieu Bertin
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