Maximizing Volume with Derivatives

The video tutorial explains how to determine the height of a suitcase that maximizes its volume, given constraints on its dimensions. It involves finding critical numbers of the volume function using derivatives and verifying the solution graphically. The tutorial demonstrates the application of calculus concepts like the product rule and chain rule to solve optimization problems.

What is the maximum allowable sum of the dimensions of the luggage?

• 90 inches

• 50 inches

• 100 inches

• 78 inches

What is the formula for the volume of the luggage?

• V = l * w * h

• V = h * (39 - 12h)

• V = 2h * (39 - 12h)

• V = h + (39 - 12h)

What mathematical method is used to find the critical numbers of the volume function?

• Integration

• Differentiation

• Algebraic manipulation

• Graphical analysis

Which rule is applied to find the derivative of the volume function?

• Sum rule

• Power rule

• Product rule

• Quotient rule

What is the derivative of the function 39 - 12h with respect to h?

• -2

• 12

• -12

• 2

What is the critical height that maximizes the volume?

• 78 inches

• 26 inches

• 12 inches

• 39 inches

What happens to the volume function at the critical height of 26 inches?

• It reaches a minimum

• It reaches a maximum

• It increases

• It decreases

What is the maximum volume achieved at the critical height?

• 20,000 cubic inches

• 15,000 cubic inches

• 17,576 cubic inches

• 10,000 cubic inches

How is the maximum volume verified graphically?

• By observing the change from increasing to decreasing

• By finding the intersection points

• By checking the slope

• By calculating the area under the curve

What does the graph of the derivative function indicate at H = 78?

• No change

• A point of inflection

• A relative maximum

• A relative minimum