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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![**Surface Area and Volume Maximization of Rectangular Solids**
**Problem Statement:**
Determine the dimensions of a rectangular solid (with a square base) with maximum volume if its surface area is 37.5 square centimeters.
**To solve:**
Fill in the following values:
- \_\_\_\_ cm (smallest value)
- \_\_\_\_ cm
- \_\_\_\_ cm (largest value)
This problem requires identifying the dimensions (length, width, and height) of a rectangular solid that maximizes its volume given a fixed surface area. The solid has a square base, which implies that the length and width are equal, and the height needs to be determined along with the length and width.
Given that the surface area (SA) is specified as 37.5 square centimeters, you will need to formulate the equations based on the surface area and volume formulas of a rectangular solid and solve for the dimensions that yield the maximum volume.
**Key Concepts:**
1. Surface Area (SA) of a rectangular solid with a square base:
\[ SA = 2a^2 + 4ah \]
2. Volume (V) of the rectangular solid:
\[ V = a^2h \]
Where:
- \( a \) = length of the sides of the square base,
- \( h \) = height of the rectangular solid.
**Steps to Solve:**
1. Express height \( h \) in terms of the side length \( a \) using the given surface area.
2. Substitute this expression into the volume formula.
3. Determine the dimensions by maximizing the volume through calculus or algebraic methods.
By calculating accordingly, you will find the required dimensions for the maximum volume under the given surface area constraint.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F08c083ee-5dc4-4392-bfaa-a5f4117ce6af%2F7df08189-bfd5-45de-bfae-aef391e08b61%2Fy48c1hf.jpeg&w=3840&q=75)
Transcribed Image Text:**Surface Area and Volume Maximization of Rectangular Solids**
**Problem Statement:**
Determine the dimensions of a rectangular solid (with a square base) with maximum volume if its surface area is 37.5 square centimeters.
**To solve:**
Fill in the following values:
- \_\_\_\_ cm (smallest value)
- \_\_\_\_ cm
- \_\_\_\_ cm (largest value)
This problem requires identifying the dimensions (length, width, and height) of a rectangular solid that maximizes its volume given a fixed surface area. The solid has a square base, which implies that the length and width are equal, and the height needs to be determined along with the length and width.
Given that the surface area (SA) is specified as 37.5 square centimeters, you will need to formulate the equations based on the surface area and volume formulas of a rectangular solid and solve for the dimensions that yield the maximum volume.
**Key Concepts:**
1. Surface Area (SA) of a rectangular solid with a square base:
\[ SA = 2a^2 + 4ah \]
2. Volume (V) of the rectangular solid:
\[ V = a^2h \]
Where:
- \( a \) = length of the sides of the square base,
- \( h \) = height of the rectangular solid.
**Steps to Solve:**
1. Express height \( h \) in terms of the side length \( a \) using the given surface area.
2. Substitute this expression into the volume formula.
3. Determine the dimensions by maximizing the volume through calculus or algebraic methods.
By calculating accordingly, you will find the required dimensions for the maximum volume under the given surface area constraint.
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