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Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Hass 15th EditionCh. 4 - Applications of Derivatives문제 4.5.20a
4장, 문제 4.5.20a

20.The U.S. Postal Service will accept a box for domestic shipment only if the sum of its length and girth (distance around) does not exceed 108 in. a.What dimensions will give a box with a square end the largest possible volume?
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Step 1: Define the variables. Let the side length of the square end be 'x' (in inches), and the length of the box be 'L' (in inches). The girth is the perimeter of the square end, which is 4x.
Step 2: Write the constraint equation. The sum of the length and girth must not exceed 108 inches. Therefore, the constraint is: L + 4x ≤ 108.
Step 3: Express the volume of the box. The volume of the box is given by the area of the square end multiplied by the length: V = x² * L.
Step 4: Solve for 'L' using the constraint equation. From the constraint, L = 108 - 4x. Substitute this into the volume equation to express volume in terms of 'x': V = x² * (108 - 4x).
Step 5: Maximize the volume. To find the dimensions that give the largest possible volume, take the derivative of V with respect to 'x', set it equal to zero, and solve for 'x'. Then, use the constraint equation to find the corresponding 'L'.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Volume of a Box

The volume of a box is calculated by multiplying its length by its width and height. In this case, since the box has a square end, if we denote the side length of the square as 'x' and the length as 'L', the volume can be expressed as V = x^2 * L. Understanding how to express volume in terms of the box's dimensions is crucial for maximizing it under given constraints.
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04:48
Finding Volume Using Disks

Girth Calculation

Girth is defined as the distance around the box, specifically calculated as 2 times the width plus 2 times the height. For a box with a square end, this simplifies to G = 4x + 2L. The problem states that the sum of the length and girth must not exceed 108 inches, which provides a constraint that must be satisfied when determining the optimal dimensions for maximum volume.
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가이드 코스
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Determining Error and Relative Error Example 1

Optimization in Calculus

Optimization involves finding the maximum or minimum values of a function subject to certain constraints. In this scenario, we need to maximize the volume function V = x^2 * L while adhering to the constraint given by the girth equation. Techniques such as substitution and taking derivatives to find critical points will be essential in solving this problem effectively.
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