Textbook Question
Turning a corner with a pole
What is the length of the longest pole that can be carried horizontally around a corner at which a corridor that is a ft wide and a corridor that is b ft wide meet at right angles?
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Ch. 4 - Applications of the Derivative
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 4, Problem 81b
{Use of Tech} Optimal boxes Imagine a lidless box with height h and a square base whose sides have length x. The box must have a volume of 125 ft³.
b. Based on your graph in part (a), estimate the value of x that produces the box with a minimum surface area.
Verified step by step guidance1
Step 1: Start by expressing the volume of the box in terms of x and h. Since the box has a square base, the volume V is given by V = x^2 * h. We know the volume is 125 ft³, so we have the equation x^2 * h = 125.
Step 2: Solve the equation from Step 1 for h in terms of x. This gives h = 125 / x^2.
Step 3: Write the formula for the surface area S of the box. Since the box is lidless, the surface area is the area of the base plus the area of the four sides. This is given by S = x^2 + 4xh.
Step 4: Substitute the expression for h from Step 2 into the surface area formula from Step 3. This gives S = x^2 + 4x(125 / x^2).
Step 5: Simplify the expression for S obtained in Step 4. This results in S = x^2 + 500/x. To find the value of x that minimizes the surface area, you would take the derivative of S with respect to x, set it to zero, and solve for x.
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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 by multiplying its height by the area of its base. For a box with a square base of side length x and height h, the volume V is given by the formula V = x²h. In this problem, the volume is fixed at 125 ft³, which establishes a relationship between x and h that must be maintained while optimizing for surface area.
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Surface Area of a Box
The surface area of a box is the total area of all its faces. For a lidless box with a square base, the surface area S can be expressed as S = x² + 4xh, where x² is the area of the base and 4xh accounts for the four vertical sides. Minimizing the surface area while maintaining a constant volume is a key aspect of this optimization problem.
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Optimization in Calculus
Optimization involves finding the maximum or minimum values of a function. In this context, we use calculus techniques such as taking derivatives to find critical points where the surface area is minimized, subject to the constraint of a fixed volume. This often involves setting up a function for surface area in terms of a single variable and applying the first and second derivative tests to identify optimal dimensions.
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Related Practice
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Textbook Question
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Textbook Question
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Textbook Question
Use limit methods to determine which of the two given functions grows faster, or state that they have comparable growth rates.
x² ln x; x³
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