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Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 4 - Applications of DerivativesProblem 4.5.17b
Chapter 4, Problem 4.5.17b

[Technology Exercise] 17. Designing a suitcase A 24-in.-by-36-in. sheet of cardboard is folded in half to form a 24-in.-by-18-in. rectangle as shown in the accompanying figure. Then four congruent squares of side length x are cut from the corners of the folded rectangle. The sheet is unfolded, and the six tabs are folded up to form a box with sides and a lid.
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b. Find the domain of V for the problem situation and graph V over this domain.

Verified step by step guidance
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Step 1: Understand the problem setup. The cardboard is folded in half to form a rectangle of dimensions 24 inches by 18 inches. Four congruent squares of side length x are cut from the corners of this folded rectangle. When unfolded, the cardboard forms a box with a base and six tabs. The volume of the box depends on the value of x.
Step 2: Write the expression for the dimensions of the box. After cutting squares of side length x, the base dimensions of the box become (24 - 2x) inches by (18 - 2x) inches. The height of the box is x inches, as the tabs are folded up to form the sides.
Step 3: Derive the formula for the volume of the box. The volume V is given by the product of the base area and the height: V = (24 - 2x)(18 - 2x)x. Expand this expression to get the volume in terms of x.
Step 4: Determine the domain of x. Since x represents the side length of the squares cut from the corners, it must satisfy the condition that the dimensions of the base remain positive. This means 24 - 2x > 0 and 18 - 2x > 0. Solve these inequalities to find the range of x values.
Step 5: Graph the volume function V over the domain. Use the derived formula for V and plot it over the valid domain of x. The graph will show how the volume changes as x varies within the domain.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Volume of a Box

The volume of a box is calculated using the formula V = length × width × height. In this problem, the dimensions of the box are influenced by the size of the squares cut from the corners, represented by the variable x. Understanding how to express the volume in terms of x is crucial for solving the problem.
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Finding Volume Using Disks

Domain of a Function

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. In this context, the domain must consider the physical constraints of the problem, such as the size of the cardboard and the requirement that x must be less than half the shorter side of the rectangle to ensure the squares can be cut.
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Finding the Domain and Range of a Graph

Graphing Functions

Graphing a function involves plotting its output values against its input values on a coordinate plane. For this problem, once the volume function V(x) is established, graphing it over the identified domain will visually represent how the volume changes with different sizes of the squares cut from the corners, aiding in understanding the relationship between x and V.
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Related Practice
Textbook Question

Finding Antiderivatives

In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.

1/(3³√x)

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Textbook Question

Identifying Extrema


In Exercises 19–40:


b. Identify the function’s local extreme values, if any, saying where they occur.


f(θ) = 3θ² − 4θ³

Textbook Question

Finding Antiderivatives

In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.

1 / 2x³

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Textbook Question

114. Parabolas

b. When is the parabola concave up? Concave down?

Textbook Question

Identifying Extrema


In Exercises 19–40:


b. Identify the function’s local extreme values, if any, saying where they occur.


f(r) = 3r³ + 16r

Textbook Question

Finding Antiderivatives

In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.

x⁻³/2 + x²

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