Chapter 8, Problem 7P

a)

Summary Introduction

To determine: The location that will yield the greatest profit if monthly demand is 200 and 300 cars respectively.

Introduction: Location is where a firm chooses to site its operations. Location decisions can large effects on expenses and incomes. Location choices are normally quite imperative to both substantial and private companies. The area choice directly affects an operation's expenses and also its capacity to serve clients.

Answer to Problem 7P

The location outside of the city will yield greatest profit if the monthly demand is 200 and location inside the city will yield greatest profit if the monthly demand is 300 cars.

Explanation of Solution

Given Information:

It is given that the fixed monthly costs of location inside the city are $7,000 and location outside the city is $4,700. The variable costs for location inside the city are $30 per car and for location outside is $40. The dealer price per car is $90. The monthly demands are 200 and 300 cars.

Calculate the total profit of location that will yield the greatest profit if the monthly demand is 200.

It is calculated by subtracting dealer price with variable cost and the result is multiplied with monthly demand and the whole result is subtracted with fixed costs.

$\begin{array}{c}\mathrm{Total}{\text{Profits}}_{\mathrm{Location}\mathrm{Cit}}=\left[\left(\left(\mathrm{Costs}-\mathrm{Variable}\mathrm{Costs}\right)\times \mathrm{Total}\mathrm{Units}\right)-\text{Fixed}\text{Costs}\right]\\ =\left[\left(\left(\$90-\$30\right)\times 200\right)-\$7,000\right]\\ =\left[\left(\$60\times 200\right)-\$7,000\right]\\ =\left[\$12,000-\$7,000\right]\\ =\$5,000\end{array}$

$\begin{array}{c}\mathrm{Total}{\text{Profits}}_{\mathrm{Location}\mathrm{Out}}=\left[\left(\left(\mathrm{Costs}-\text{Variable}\text{Costs}\right)\times \mathrm{Total}\mathrm{Units}\right)-\text{Fixed}\text{Costs}\right]\\ =\left[\left(\left(\$90-\$40\right)\times 200\right)-\$4,700\right]\\ =\left[\left(\$50\times 200\right)-\$4,700\right]\\ =\left[\$10,000-\$4,700\right]\\ =\$5,300\end{array}$

From the results obtained Location Out yield the greatest profit if the monthly demand is 200 with $5,300 compared to Location Cit which is $5,000.

Hence location that will yield the greatest profit if the monthly demand is 200 is Location Out.

Calculate the total profit of location that will yield the greatest profit if the monthly demand is 300.

Given Information:

It is given that the fixed monthly costs of Location Cit are $7,000 and Location Out is $4,700. The variable costs for Location Cit are $30 per car and Location Out is $40. The dealer price per car is $90. The monthly demands are 200 and 300 cars.

It is calculated by subtracting dealer price with variable cost and the result is multiplied with monthly demand and the whole result is subtracted with fixed costs.

$\begin{array}{c}\mathrm{Total}{\text{Profits}}_{\mathrm{Location}\mathrm{Cit}}=\left[\left(\left(\mathrm{Costs}-\text{Variable}\text{Cost}s\right)\times \mathrm{Total}\mathrm{Units}\right)-\text{Fixed}\text{Costs}\right]\\ =\left[\left(\left(\$90-\$30\right)\times 300\right)-\$7,000\right]\\ =\left[\left(\$60\times 300\right)-\$7,000\right]\\ =\left[\$18,000-\$7,000\right]\\ =\$11,000\end{array}$

$\begin{array}{c}\mathrm{Total}{\text{Profits}}_{\mathrm{Location}\mathrm{Out}}=\left[\left(\left(\mathrm{Costs}-\mathrm{Variable}\mathrm{Costs}\right)\times \mathrm{Total}\mathrm{Units}\right)-\mathrm{Fixed}\mathrm{Costs}\right]\\ =\left[\left(\left(\$90-\$40\right)\times 300\right)-\$4,700\right]\\ =\left[\left(\$50\times 300\right)-\$4,700\right]\\ =\left[\$15,000-\$4,700\right]\\ =\$10,300\end{array}$

From the results obtained location city yield the greatest profit if the monthly demand is 300 with $10,300 compared to location city which is $11,000.

Hence, location that will yield the greatest profit if the monthly demand is 300 is location city.

b)

Summary Introduction

To determine: The volume of output with the two locations yield the same monthly profit.

Introduction: Location is where a firm chooses to site its operations. Location decisions can large effects on expenses and incomes. Location choices are normally quite imperative to both substantial and private companies. The area choice directly affects an operation's expenses and also its capacity to serve clients.

Answer to Problem 7P

The volume of output with the two locations yield the same monthly profit is 230 cars.

Explanation of Solution

Given Information:

It is given that the fixed monthly costs of Location City are $7,000 and Location Out is $4,700. The variable costs for Location Cit are $30 per car and Location Out is $40. The dealer price per car is $90. The monthly demands are 200 and 300 cars.

Calculate the volume of output with the two locations yield the same monthly profit.

$\begin{array}{c}\mathrm{Total}{\text{Profits}}_{\mathrm{Location}\mathrm{Cit}}=\left[\left(\left(\mathrm{Costs}-\mathrm{Variable}\mathrm{Costs}\right)\times \mathrm{Total}\mathrm{Units}\right)-\text{Fixed}\text{Costs}\right]\\ =\left[\left(\left(\$90-\$30\right)\times \mathrm{Total}\mathrm{Units}\right)-\$7,000\right]\\ \mathrm{Total}{\text{Profits}}_{\mathrm{Location}\mathrm{Out}}=\left[\left(\left(\text{Costs}-\mathrm{Variable}\text{Costs}\right)\times \mathrm{Total}\mathrm{Units}\right)-\text{Fixed}\text{Costs}\right]\\ =\left[\left(\left(\$90-\$40\right)\times \mathrm{Total}\mathrm{Units}\right)-\$4,700\right]\end{array}$

The total units are denoted as Q. Solving for Q,

$\begin{array}{c}\left[\left(\text{Q}\times \left(\$90-\$30\right)\right)-\$7,000\right]=\left[\left(\text{Q}\times \left(\$90-\$40\right)\right)-\$4,700\right]\\ \$60\text{Q}-\$7,000=\$50\text{Q}-\$4,700\\ \$60\text{Q}-\$50\text{Q}=-\$4,700-\left(-\$7,000\right)\\ \$10\text{Q}=\$2,300\\ \text{Q}=\left[\frac{\$2,300}{\$10}\right]\\ =230\mathrm{units}\end{array}$

Hence, the volume of output with the two locations yield the same monthly profit is 230 cars.