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.

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 $35$: \[ x = \frac{4760}{35} = 136 \] 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 $ \boxed{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.

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