Container Volume and Cost Analysis

The video tutorial addresses a maximization problem involving a rectangular storage container with an open top. The container's volume is fixed at 10 cubic meters, and the base length is twice its width. The tutorial guides through calculating the height in terms of the width, deriving a cost function for the materials, and using calculus to find the minimum cost for constructing the container. The solution is verified by checking the critical points and confirming the minimum cost.

What is the volume of the rectangular storage container?

• 5 meters cubed

• 10 meters cubed

• 15 meters cubed

• 20 meters cubed

If the width of the base is x, what is the length of the base?

• 2x

• 3x

• 4x

• x

How is the height of the container expressed in terms of x?

• 10x^2

• 10x^-2

• 5x^-2

• 5x^2

What is the cost per square meter for the base material?

• $5

• $6

• $12

• $10

What is the cost per square meter for the side material?

• $6

• $10

• $12

• $5

What is the simplified expression for the cost function?

• 20x^2 + 180x^-1

• 30x^2 + 150x^-1

• 40x^2 + 160x^-1

• 50x^2 + 140x^-1

What is the derivative of the cost function?

• 20x + 90x^-2

• 40x - 180x^-2

• 40x + 180x^-2

• 20x - 90x^-2

What is the critical point found for x?

• 4.65

• 1.65

• 2.65

• 3.65

What is the method used to confirm if the critical point is a minimum?

• Trial and error

• Graphical method

• Second derivative test

• First derivative test

What is the final value of x for the cheapest container?

• 1.75

• 1.55

• 1.85

• 1.65