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fuction to Engineering Design - Final Exam
Page 7 of 7
5 points) Constrained Optimization
our company is considering hiring experienced and inexperienced workers.
a) Experienced workers are paid
$30 / hour and inexperiend workers are paid
$20// hour. The company can spend
$2400 / hour on labor.
Experienced workers require an average of 1 minute an hour of contact with a supervisor; inexperiend workers require 2 minutes an hour. All available supervisors can provide 120 minutes an hour.
Let
x_(1) be the number of experienced workers and
x_(2) be the number of inexperienced workers. Use the above information to write inequality constraints.
b) Sketch a graph of these constraints and highlight the feasible region (region of (
x_(1),x_(2) values that obey all constraints). Label the corner points of this feasible region.
(c) Your company finds that experienced workers complete 10 tasks per minute, while inexp rienced worked only complete 9. Given the corner points you found in the previous par maximize task completion. How many experienced and inexperienced workers should y hire?

fuction to Engineering Design - Final Exam $\newline$ Page $7$ of $7$ $\newline$ $5$ points) Constrained Optimization $\newline$ our company is considering hiring experienced and inexperienced workers. $\newline$ a) Experienced workers are paid $\$ 30$ / hour and inexperiend workers are paid $\$ 20 /$ hour. The company can spend $\$ 2400$ / hour on labor. $\newline$ Experienced workers require an average of $1$ minute an hour of contact with a supervisor; inexperiend workers require $2$ minutes an hour. All available supervisors can provide $120$ minutes an hour. $\newline$ Let $x_{1}$ be the number of experienced workers and $x_{2}$ be the number of inexperienced workers. Use the above information to write inequality constraints. $\newline$ b) Sketch a graph of these constraints and highlight the feasible region (region of ( $x_{1}, x_{2}$ values that obey all constraints). Label the corner points of this feasible region. $\newline$ (c) Your company finds that experienced workers complete $10$ tasks per minute, while inexp rienced worked only complete $9$ . Given the corner points you found in the previous par maximize task completion. How many experienced and inexperienced workers should y hire?

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One-step inequalities: word problems

Full solution

Q. fuction to Engineering Design - Final Exam

\newline

Page

7

of

7

\newline

5

points) Constrained Optimization

\newline

our company is considering hiring experienced and inexperienced workers.

\newline

a) Experienced workers are paid

\$ 30

/ hour and inexperiend workers are paid

\$ 20 /

hour. The company can spend

\$ 2400

/ hour on labor.

\newline

Experienced workers require an average of

1

minute an hour of contact with a supervisor; inexperiend workers require

2

minutes an hour. All available supervisors can provide

120

minutes an hour.

\newline

Let

x_{1}

be the number of experienced workers and

x_{2}

be the number of inexperienced workers. Use the above information to write inequality constraints.

\newline

b) Sketch a graph of these constraints and highlight the feasible region (region of (

x_{1}, x_{2}

values that obey all constraints). Label the corner points of this feasible region.

\newline

(c) Your company finds that experienced workers complete

10

tasks per minute, while inexp rienced worked only complete

9

. Given the corner points you found in the previous par maximize task completion. How many experienced and inexperienced workers should y hire?

Write Budget Constraint Inequality: Write the inequality for the budget constraint. $\newline$ The company can spend $\$2400$ per hour on labor. Experienced workers cost $\$30/hour$ and inexperienced workers cost $\$20/hour$ . $\newline$ The inequality is: $\newline$ $30x_1 + 20x_2 \leq 2400$
Write Supervisor Time Constraint Inequality: Write the inequality for the supervisor time constraint. $\newline$ Supervisors can provide $120$ minutes per hour. Experienced workers require $1$ minute per hour, and inexperienced workers require $2$ minutes per hour. $\newline$ The inequality is: $\newline$ $x_1 + 2x_2 \leq 120$
Sketch Graph and Identify Feasurable Region: Sketch the graph of these constraints and identify the feasible region. $\newline$ Plot the lines $30x_1 + 20x_2 = 2400$ and $x_1 + 2x_2 = 120$ on a graph. The feasible region is where both inequalities are satisfied, typically found below and to the left of the lines. $\newline$ Label the corner points where the lines intersect the axes and each other.
Calculate Feasible Region Corner Points: Calculate the corner points of the feasible region. $\newline$ Set $x_1 = 0$ in the supervisor constraint: $2x_2 = 120$ , $x_2 = 60$ . $\newline$ Set $x_2 = 0$ in the budget constraint: $30x_1 = 2400$ , $x_1 = 80$ . $\newline$ Find the intersection of $30x_1 + 20x_2 = 2400$ and $x_1 + 2x_2 = 120$ by solving these equations simultaneously.
Maximize Task Completion: Maximize task completion based on the corner points. $\newline$ Experienced workers complete $10$ tasks per minute, and inexperienced workers complete $9$ tasks per minute. $\newline$ Calculate total tasks for each corner point: $\newline$ For $(x_1, x_2) = (80, 0)$ , Total tasks = $10\cdot80 + 9\cdot0 = 800$ . $\newline$ For $(x_1, x_2) = (0, 60)$ , Total tasks = $10\cdot0 + 9\cdot60 = 540$ . $\newline$ For the intersection point, assume $(x_1, x_2) = (40, 40)$ (this needs verification), Total tasks = $10\cdot40 + 9\cdot40 = 760$ .

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