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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![### Calculus Problem: Finding the Time of Maximum Increase in Annual Sales
#### Problem Statement
Suppose the annual sales \( S \) of a new product is given by the function:
\[ S = \frac{1500t^2}{5 + t^2} \]
where \( t \) is the time in years, and \( 0 \leq t \leq 3 \).
**Objective:** Find the exact time when the annual sales are increasing at the greatest rate. Round your answer to three decimal places.
This problem involves finding the maximum rate of increase in sales over the given period. We will utilize calculus to determine the exact time.
#### Solution Steps:
1. **Find the first derivative of \( S \) with respect to \( t \) to determine the rate of change of sales.**
2. **Identify critical points by setting the first derivative equal to zero and solving for \( t \).**
3. **Analyze the critical points to determine which one corresponds to the maximum rate of increase.**
4. **Verify the result and round the answer to three decimal places.**
Please follow the detailed solution approach to grasp the underlying principles and methods used in this optimization problem.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7c1f21d3-5711-44a9-a68b-ddf542eb9bce%2F3668eb6c-7ac9-4b3d-a713-671036e0be14%2Fws3uuymf.png&w=3840&q=75)
Transcribed Image Text:### Calculus Problem: Finding the Time of Maximum Increase in Annual Sales
#### Problem Statement
Suppose the annual sales \( S \) of a new product is given by the function:
\[ S = \frac{1500t^2}{5 + t^2} \]
where \( t \) is the time in years, and \( 0 \leq t \leq 3 \).
**Objective:** Find the exact time when the annual sales are increasing at the greatest rate. Round your answer to three decimal places.
This problem involves finding the maximum rate of increase in sales over the given period. We will utilize calculus to determine the exact time.
#### Solution Steps:
1. **Find the first derivative of \( S \) with respect to \( t \) to determine the rate of change of sales.**
2. **Identify critical points by setting the first derivative equal to zero and solving for \( t \).**
3. **Analyze the critical points to determine which one corresponds to the maximum rate of increase.**
4. **Verify the result and round the answer to three decimal places.**
Please follow the detailed solution approach to grasp the underlying principles and methods used in this optimization problem.
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