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Maximizing Volume and Minimizing Paper Use: Open Box and Print Dimensions in AP Calculus

In this AP Calculus AB problem set, we explore two optimization scenarios. First, we determine the dimensions of an open box with a square base and a surface area of 108 square inches, aiming to maximize its volume. Next, we calculate the optimal dimensions of a rectangular print area that contains 24 square inches of content while minimizing paper usage, considering specific margin requirements. Step-by-step analysis includes identifying primary and secondary intervals, testing values, and deriving maximum and minimum dimensions.

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Maximizing Volume and Minimizing Paper Use: Open Box and Print Dimensions in AP Calculus

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  1. AP Calculus AB Day 12 Section 3.7 Perkins

  2. 1. An open box having a square base and a surface area of 108 square inches is to have a maximum volume. Find its dimensions. Primary Secondary Domain of x will range from x being as small as possible to x as large as possible. Largest (y is near zero) Smallest (x is near zero) Intervals: Test values: V ’(test pt) V(x) rel max Dimensions: 6 in x 6 in x 3 in

  3. 2. A rectangular page is to contain 24 square inches of print. The margins at the top and bottom are 1.5 inches. The margins on each side are 1 inch. What should the dimensions of the print be to use the least paper? Primary Secondary Largest (y is near zero) Smallest (x is near zero) Intervals: Test values: Print dimensions: 6 in x 4 in A ’(test pt) A(x) Page dimensions: 9 in x 6 in rel min

  4. AP Calculus AB Day 12 Section 3.7 Perkins

  5. 1. An open box having a square base and a surface area of 108 square inches is to have a maximum volume. Find its dimensions.

  6. 2. A rectangular page is to contain 24 square inches of print. The margins at the top and bottom are 1.5 inches. The margins on each side are 1 inch. What should the dimensions of the print be to use the least paper?

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