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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
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:** 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 \( E(x) \).
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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