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A box has a square base of side x and height h. (a) Find the dimensions x, h for which the volume is 25 and the surface area is as small as possible. (Use symbolic notation and fractions where needed.) 2.92 Incorrect h = 2.92

A box has a square base of side x and height h. (a) Find the dimensions x, h for which the volume is 25 and the surface area is as small as possible. (Use symbolic notation and fractions where needed.) 2.92 Incorrect h = 2.92

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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## Problem Statement A box has a square base of side \( x \) and height \( h \). ### Part (a) Find the dimensions \( x, h \) for which the volume is 25 and the surface area is as small as possible. *(Use symbolic notation and fractions where needed.)* **Input:** \( x = 2.92 \) - Incorrect \( h = 2.92 \) - Incorrect ### Part (b) Find the dimensions \( x, h \) for which the surface area is 23 and the volume is as large as possible. *(Use symbolic notation and fractions where needed.)* **Input:** \( x = 1.96 \) - Incorrect \( h = 1.96 \) - Incorrect ## Explanation For each part, the task is to determine the dimensions of a box (side length \( x \) and height \( h \)) according to given constraints: - **Part (a):** The volume is fixed at 25 units. The goal is to minimize the surface area. - **Part (b):** The surface area is fixed at 23 units. The goal is to maximize the volume. In both parts, the attempted solutions were incorrect. This suggests a need for recalculation or revisiting the formulas used to obtain these dimensions. Students should consider revisiting the relationships between volume \( V = x^2 \cdot h \) and surface area \( A = 2x^2 + 4xh \) to find the correct dimensions. Consider utilizing optimization techniques, such as taking derivatives and setting them equal to zero to find minimum and maximum values, respectively.
Transcribed Image Text:## Problem Statement A box has a square base of side \( x \) and height \( h \). ### Part (a) Find the dimensions \( x, h \) for which the volume is 25 and the surface area is as small as possible. *(Use symbolic notation and fractions where needed.)* **Input:** \( x = 2.92 \) - Incorrect \( h = 2.92 \) - Incorrect ### Part (b) Find the dimensions \( x, h \) for which the surface area is 23 and the volume is as large as possible. *(Use symbolic notation and fractions where needed.)* **Input:** \( x = 1.96 \) - Incorrect \( h = 1.96 \) - Incorrect ## Explanation For each part, the task is to determine the dimensions of a box (side length \( x \) and height \( h \)) according to given constraints: - **Part (a):** The volume is fixed at 25 units. The goal is to minimize the surface area. - **Part (b):** The surface area is fixed at 23 units. The goal is to maximize the volume. In both parts, the attempted solutions were incorrect. This suggests a need for recalculation or revisiting the formulas used to obtain these dimensions. Students should consider revisiting the relationships between volume \( V = x^2 \cdot h \) and surface area \( A = 2x^2 + 4xh \) to find the correct dimensions. Consider utilizing optimization techniques, such as taking derivatives and setting them equal to zero to find minimum and maximum values, respectively.
**Question 11 of 14** Use a computer algebra system to find the transition points. \[ y = 5e^{-x^2} \ln(x) \] (Give your answers to six decimal places. Provide your answers as comma separated x-values. Enter DNE if a point does not exist.) 1. **Find the local maxima.** \( x = \) [Input field] 2. **Find the local minima.** \( x = \) [Input field] 3. **Find the inflection points.** \( x = \) [Input field]
Transcribed Image Text:**Question 11 of 14** Use a computer algebra system to find the transition points. \[ y = 5e^{-x^2} \ln(x) \] (Give your answers to six decimal places. Provide your answers as comma separated x-values. Enter DNE if a point does not exist.) 1. **Find the local maxima.** \( x = \) [Input field] 2. **Find the local minima.** \( x = \) [Input field] 3. **Find the inflection points.** \( x = \) [Input field]
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