Questions: Your firm is considering adding a product line. The accounting department has determined that the cost function for this new product will be C(x)=325x+4760 and the revenue function will be R(x)=360x, where x is the number of units sold. Additionally, the sales department has determined that you reasonably can expect to sell around 173 units. You must decide whether to go ahead with the new product line.
Find the break-even quantity:
Find the profit function: P(x)=
Make your decision:
Proceed with the product line: the products are profitable and we can reasonably expect to meet our break-even quantity.
Cancel the product line: the quantity needed to break even is more than our firm can reasonably expect to sell.
Cancel the product line: producing these products is not profitable.
Transcript text: Your firm is considering adding a product line. The accounting department has determined that the cost function for this new product will be $C(x)=325 x+4760$ and the revenue function will be $R(x)=360 x$, where $x$ is the number of units sold. Additionally, the sales department has determined that you reasonably can expect to sell around 173 units. You must decide whether to go ahead with the new product line.
Find the break-even quantity: $\square$
Find the profit function: $P(x)=$ $\square$
Make your decision:
Proceed with the product line: the products are profitable and we can reasonably expect to meet our break-even quantity.
Cancel the product line: the quantity needed to break even is more than our firm can reasonably expect to sell.
Cancel the product line: producing these products is not profitable.
Solution
Solution Steps
Step 1: Find the break-even quantity
To find the break-even quantity, set the cost function equal to the revenue function:
C(x)=R(x)325x+4760=360x
Subtract 325x from both sides:
4760=35x
Divide both sides by 35:
x=354760=136
The break-even quantity is 136 units.
Step 2: Find the profit function
The profit function P(x) is the difference between the revenue function and the cost function:
P(x)=R(x)−C(x)
Substitute the given functions:
P(x)=360x−(325x+4760)
Simplify:
P(x)=35x−4760
Step 3: Make your decision
The break-even quantity is 136 units, and the sales department expects to sell around 173 units. Since 173>136, the firm can reasonably expect to meet the break-even quantity. Additionally, the profit function P(x)=35x−4760 shows that profit increases as x increases beyond the break-even point. Therefore, the decision is:
Proceed with the product line: the products are profitable and we can reasonably expect to meet our break-even quantity.
Final Answer
The break-even quantity is 136.
The profit function is P(x)=35x−4760.
Proceed with the product line: the products are profitable and we can reasonably expect to meet our break-even quantity.