6) Bob makes $45 an hour tutoning for 10 houws a week. He wants to raise his hourly rate, but ifhe does, he will love I hour oA wark for every $3 increase to his howrly mate. Write an egueetion to model how bob Can maximize his rate while ninimiz ing his houls of work.
6) Bob makes $45 an hour tutoning for 10 houws a week. He wants to raise his hourly rate, but ifhe does, he will love I hour oA wark for every $3 increase to his howrly mate. Write an egueetion to model how bob Can maximize his rate while ninimiz ing his houls of work.
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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Question
![**Problem Statement:**
Bob makes $45 an hour tutoring for 10 hours a week. He wants to raise his hourly rate, but if he does, he will lose 1 hour of work for every $3 increase to his hourly rate. Write an equation to model how Bob can maximize his rate while minimizing his hours of work.
---
**Solution Explanation:**
To create a model for Bob's situation, we first define the variables:
- Let \( x \) be the number of $3 increases in his hourly rate.
- Bob's new hourly rate will be \( 45 + 3x \).
- The number of hours he works per week will be \( 10 - x \).
**Equation for Total Earnings:**
Bob's total earnings in a week can be modeled by the equation:
\[
E(x) = (45 + 3x)(10 - x)
\]
In this equation:
- \( 45 + 3x \) represents his increased hourly rate.
- \( 10 - x \) represents the reduced hours worked based on his rate increase.
The goal is to determine the value of \( x \) that maximizes Bob's weekly earnings \( E(x) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3212682e-3570-42bd-a59e-3d66eb6146ee%2F6f064290-c40b-48a4-93cd-cc25d451b42b%2F5o2fvbr_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Bob makes $45 an hour tutoring for 10 hours a week. He wants to raise his hourly rate, but if he does, he will lose 1 hour of work for every $3 increase to his hourly rate. Write an equation to model how Bob can maximize his rate while minimizing his hours of work.
---
**Solution Explanation:**
To create a model for Bob's situation, we first define the variables:
- Let \( x \) be the number of $3 increases in his hourly rate.
- Bob's new hourly rate will be \( 45 + 3x \).
- The number of hours he works per week will be \( 10 - x \).
**Equation for Total Earnings:**
Bob's total earnings in a week can be modeled by the equation:
\[
E(x) = (45 + 3x)(10 - x)
\]
In this equation:
- \( 45 + 3x \) represents his increased hourly rate.
- \( 10 - x \) represents the reduced hours worked based on his rate increase.
The goal is to determine the value of \( x \) that maximizes Bob's weekly earnings \( E(x) \).
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