Questions: 2. A manufacturer can sell a certain product for 110 per unit. Total cost consists of a fixed overhead of 7,500 plus production cost of 60 per unit. (a) How many units must the manufacturer sell to break even? (b) What is the manufacturer's profit or loss if 100 units are sold? (c) How many units must be sold for the manufacturer to make a profit of 1250?

2. A manufacturer can sell a certain product for 110 per unit. Total cost consists of a fixed overhead of 7,500 plus production cost of 60 per unit.

(a) How many units must the manufacturer sell to break even?

(b) What is the manufacturer's profit or loss if 100 units are sold?

(c) How many units must be sold for the manufacturer to make a profit of 1250?

Transcript text: 2. A manufacturer can sell a certain product for $\$ 110$ per unit. Total cost consists of a fixed overhead of $\$ 7,500$ plus production cost of $\$ 60$ per unit. [3] (a) How many units must the manufacturer sell to break even? [3] (b) What is the manufacturer's profit or loss if 100 units are sold? [3] (c) How many units must be sold for the manufacturer to make a profit of $\$ 1250$ ?

Solution

Solution Steps

Solution Approach

(a) To find the break-even point, calculate the number of units where total revenue equals total cost. Set the revenue equation equal to the cost equation and solve for the number of units.

(b) To find the profit or loss for selling 100 units, calculate the total revenue and total cost for 100 units and subtract the total cost from the total revenue.

(c) To find the number of units needed to achieve a profit of $1250, set the profit equation (total revenue minus total cost) equal to $1250 and solve for the number of units.

Step 1: Determine the Break-even Point

To find the break-even point, we set the total revenue equal to the total cost. The revenue from selling $ x $ units is given by $ 110x $, and the total cost is given by $ 7500 + 60x $. Setting these equal gives:

\[ 110x = 7500 + 60x \]

Solving for $ x $, we have:

\[ 110x - 60x = 7500 \\ 50x = 7500 \\ x = \frac{7500}{50} = 150 \]

Thus, the manufacturer must sell $ \boxed{150} $ units to break even.

Step 2: Calculate Profit or Loss for 100 Units

To find the profit or loss when 100 units are sold, we calculate the total revenue and total cost for 100 units. The total revenue is $ 110 \times 100 = 11000 $, and the total cost is $ 7500 + 60 \times 100 = 13500 $. The profit or loss is:

\[ \text{Profit/Loss} = 11000 - 13500 = -2500 \]

Thus, the manufacturer incurs a loss of $ \boxed{-2500} $.

Step 3: Determine Units for a Profit of $1250

To find the number of units needed to achieve a profit of $1250, we set the profit equation equal to 1250. The profit is given by:

\[ 110x - (7500 + 60x) = 1250 \]

Simplifying, we have:

\[ 50x - 7500 = 1250 \\ 50x = 8750 \\ x = \frac{8750}{50} = 175 \]

Thus, the manufacturer must sell $ \boxed{175} $ units to achieve a profit of $1250.