Question
Profit, Revenue, Costs
Maria paints custom tote bags and sells them at a community fair. She uses the equation
P = 12n − (3.5n + 40)
to calculate her total profit, in dollars, when she sells n tote bags.
Based on this equation, how much does she charge for each tote bag?
Maria paints custom tote bags and sells them at a community fair. She uses the equation
P = 12n − (3.5n + 40)
to calculate her total profit, in dollars, when she sells n tote bags.
Based on this equation, how much does she charge for each tote bag?
Answers
GPT-4o mini
To analyze Maria's profit equation, we start by simplifying it:
\[
P = 12n - (3.5n + 40)
\]
Distributing the negative sign gives:
\[
P = 12n - 3.5n - 40
\]
Now, we combine like terms:
\[
P = (12n - 3.5n) - 40
\]
This simplifies to:
\[
P = 8.5n - 40
\]
In this equation, \( P \) represents profit, and \( n \) is the number of tote bags sold. The term \( 8.5n \) reflects the profit from selling \( n \) tote bags, while the fixed cost \( -40 \) represents her expenses.
The total revenue from selling \( n \) tote bags can be expressed as the price per bag multiplied by the number of bags sold. Let \( P_b \) be the price Maria charges for each tote bag. Therefore, her total revenue \( R \) can be represented as:
\[
R = P_b \cdot n
\]
According to her profit calculation, profit can also be expressed as:
\[
P = R - C
\]
where \( C \) denotes costs. We know from the profit equation that:
\[
R = C + P
\]
To isolate the price per bag, we note that revenue must account for both costs and profit. Given that the costs related to each bag are included in the profit equation and observed as \( 3.5n + 40 \), we can find out how much revenue she aims for after covering these costs and expenses.
The revenue contributed by selling \( n \) tote bags, from her profit equation, can be set against the costs. Since we only seek the sale price per bag, we observe the form of the profit equation collapsed down. Since revenue minus costs generates her profit, we have that:
\[
R = 12n
\]
Where \( 12n \) illustrates her total intake from the sales. When \( n = 1 \):
\[
R = 12 \text{ dollars}
\]
Hence, the price per tote bag \( P_b \) is:
\[
P_b = 12 \text{ dollars}
\]
Thus, she charges **$12** for each tote bag.
\[
P = 12n - (3.5n + 40)
\]
Distributing the negative sign gives:
\[
P = 12n - 3.5n - 40
\]
Now, we combine like terms:
\[
P = (12n - 3.5n) - 40
\]
This simplifies to:
\[
P = 8.5n - 40
\]
In this equation, \( P \) represents profit, and \( n \) is the number of tote bags sold. The term \( 8.5n \) reflects the profit from selling \( n \) tote bags, while the fixed cost \( -40 \) represents her expenses.
The total revenue from selling \( n \) tote bags can be expressed as the price per bag multiplied by the number of bags sold. Let \( P_b \) be the price Maria charges for each tote bag. Therefore, her total revenue \( R \) can be represented as:
\[
R = P_b \cdot n
\]
According to her profit calculation, profit can also be expressed as:
\[
P = R - C
\]
where \( C \) denotes costs. We know from the profit equation that:
\[
R = C + P
\]
To isolate the price per bag, we note that revenue must account for both costs and profit. Given that the costs related to each bag are included in the profit equation and observed as \( 3.5n + 40 \), we can find out how much revenue she aims for after covering these costs and expenses.
The revenue contributed by selling \( n \) tote bags, from her profit equation, can be set against the costs. Since we only seek the sale price per bag, we observe the form of the profit equation collapsed down. Since revenue minus costs generates her profit, we have that:
\[
R = 12n
\]
Where \( 12n \) illustrates her total intake from the sales. When \( n = 1 \):
\[
R = 12 \text{ dollars}
\]
Hence, the price per tote bag \( P_b \) is:
\[
P_b = 12 \text{ dollars}
\]
Thus, she charges **$12** for each tote bag.