Maximizing the Area of a Rectangle Bounded by a Semicircle

The video tutorial explains how to determine the dimensions of a rectangle under a semicircle that maximizes its area. It involves defining the rectangle's dimensions, using calculus to find the derivative, and simplifying it to find critical points. The tutorial concludes by determining the maximum area and dimensions of the rectangle.

What is the equation of the semicircle that bounds the rectangle?

• y = sqrt(16 - x^2)

• y = 16 + x^2

• y = sqrt(16 + x^2)

• y = 16 - x^2

How is the area of the rectangle expressed in terms of x?

• 2x * sqrt(16 - x^2)

• x * sqrt(16 - x^2)

• 2x * (16 - x^2)

• x * (16 - x^2)

Which mathematical rule is applied to find the derivative of the area function?

• Difference Rule

• Product Rule

• Quotient Rule

• Sum Rule

What is the critical number for x that maximizes the area of the rectangle?

• x = 2

• x = 2 sqrt(2)

• x = sqrt(8)

• x = 4

Why is only the positive critical number considered for maximizing the area?

• Positive x is always larger

• Negative x would result in a negative area

• Negative x is not possible in this context

• Negative x is not a real number

What is the approximate decimal value of 2 sqrt(2)?

• 3.2

• 3.0

• 2.8

• 2.5

What are the dimensions of the rectangle that maximize the area?

• 8 by 4

• 4 sqrt(2) by 2 sqrt(2)

• 2 sqrt(2) by 2 sqrt(2)

• 4 by 2

What is the maximum area of the rectangle?

• 32 square units

• 12 square units

• 8 square units

• 16 square units

Which method is used to verify the maximum area graphically?

• First Derivative Test

• Second Derivative Test

• Graphical Analysis

• Numerical Approximation

What is the significance of the point x = 2 sqrt(2) in the context of this problem?

• It is where the function is not continuous

• It is where the area is minimized

• It is where the area is maximized

• It is where the derivative is undefined