Maximizing the Area of an Inscribed Rectangle

The video tutorial explains how to find the dimensions of a rectangle inscribed in a parabola with its base on the x-axis, aiming to maximize its area. The process involves setting up an equation for the area, finding critical numbers using derivatives, and verifying the maximum area using derivative tests. The final dimensions of the rectangle are calculated, and the maximum area is determined to be approximately 20.78 square units.

What is the equation of the parabola in which the rectangle is inscribed?

• y = -9 + x^2

• y = 9 + x^2

• y = 9 - x^2

• y = x^2 - 9

How is the total length of the rectangle along the x-axis expressed in terms of x?

• x

• 9 - x^2

• 2x

• x^2

What is the expression for the area of the rectangle in terms of x?

• 18x - 2x^3

• 9x - x^3

• 2x(9 - x^2)

• x(9 - x^2)

What is the derivative of the area function, A'(x)?

• 18x - 6x

• 18 - 6x^2

• 9 - 2x^2

• 18x - 2x^3

What value of x gives the critical point for the area function?

• x = 3

• x = 0

• x = sqrt(3)

• x = -sqrt(3)

Which test is used to verify that the critical point is a maximum?

• Second Derivative Test

• Concavity Test

• First Derivative Test

• Limit Test

What is the second derivative of the area function, A''(x)?

• 6x

• 12x

• -12x

• -6x

What is the length of the rectangle with the maximum area?

• 2 units

• 9 units

• 6 units

• 3 units

What is the width of the rectangle with the maximum area?

• 3 units

• sqrt(3) units

• 2 sqrt(3) units

• 2 units

What is the approximate maximum area of the rectangle?

• 15 square units

• 18 square units

• 20.78 square units

• 12 square units