Question
The owner of Comix Corner finds that he sells fewer comic books when he raises the price. When he charges p dollars for each comic book, he can expect to sell 500 - 25p comic books per month. This means his monthly revenue is p(500-25p) dollars. Each comic book costs him $4, and he pays $1,200 in other monthly expenses, so his monthly costs are 4(50025p) + 1,200 dollars. Therefore, when the owner charges p dollars for each comic book, his monthly profit in dollars is p(500-25p) (4(50025p) + 1,200). This expression can be simplified to -25p² + 600p 3,200, and then written in factored form as-25(p8)(p16). What do the numbers 8 and 16 represent in the expression? the number of comic books the owner can expect to sell, and the monthly profit in dollars, if he charges $12 for each comic book
the price per comic book to maximize the owner's monthly profit, and the number of comic books he can sell at that price
the prices the owner can charge to exactly break even
the owner's minimum and maximum monthly profit in dollars
the price per comic book to maximize the owner's monthly profit, and the number of comic books he can sell at that price
the prices the owner can charge to exactly break even
the owner's minimum and maximum monthly profit in dollars
Answers
GPT-4o mini
To analyze the factored form of the profit expression and what the numbers 8 and 16 represent, let's first clarify the profit function derived from the revenue and cost equations.
Given:
- The expected number of comic books sold when priced at \( p \) dollars each is \( 500 - 25p \).
- Monthly revenue is \( p(500 - 25p) \).
- Each comic costs $4 to produce, and there are additional monthly expenses of $1,200.
- Monthly costs are \( 4(500 - 25p) + 1,200 \).
The monthly profit \( P \) can be calculated as:
\[
P = \text{Revenue} - \text{Cost} = p(500 - 25p) - \left(4(500 - 25p) + 1,200\right)
\]
After simplifying the profit expression, it is stated that:
\[
P = -25p^2 + 600p - 3,200
\]
Factoring it leads to:
\[
P = -25(p - 8)(p - 16)
\]
In this factored form, the numbers 8 and 16 represent:
1. The prices at which the owner exactly breaks even (i.e., \( P = 0 \)). The break-even prices can be found by setting the profit equation to zero:
\[
p - 8 = 0 \quad \text{or} \quad p - 16 = 0
\]
Thus, \( p = 8 \) and \( p = 16 \) represent points where the profit is zero.
2. Therefore, the answer to what the numbers 8 and 16 represent is:
**the prices the owner can charge to exactly break even.**
Given:
- The expected number of comic books sold when priced at \( p \) dollars each is \( 500 - 25p \).
- Monthly revenue is \( p(500 - 25p) \).
- Each comic costs $4 to produce, and there are additional monthly expenses of $1,200.
- Monthly costs are \( 4(500 - 25p) + 1,200 \).
The monthly profit \( P \) can be calculated as:
\[
P = \text{Revenue} - \text{Cost} = p(500 - 25p) - \left(4(500 - 25p) + 1,200\right)
\]
After simplifying the profit expression, it is stated that:
\[
P = -25p^2 + 600p - 3,200
\]
Factoring it leads to:
\[
P = -25(p - 8)(p - 16)
\]
In this factored form, the numbers 8 and 16 represent:
1. The prices at which the owner exactly breaks even (i.e., \( P = 0 \)). The break-even prices can be found by setting the profit equation to zero:
\[
p - 8 = 0 \quad \text{or} \quad p - 16 = 0
\]
Thus, \( p = 8 \) and \( p = 16 \) represent points where the profit is zero.
2. Therefore, the answer to what the numbers 8 and 16 represent is:
**the prices the owner can charge to exactly break even.**