To find the revenue function, we need to consider both the change in the number of members and the change in the membership fee as we increase the fee by increments of [tex]$5.
1. Determine the initial conditions:
- Initial number of members: \(320\)
- Initial membership fee: \(\$[/tex]45\)
2. Understand the impact of increasing fee:
- Each time the fee increases by [tex]\(\$5\)[/tex], 10 members leave.
- Let [tex]\(x\)[/tex] denote the number of increments of [tex]\(\$5\)[/tex].
3. Define the changed values in terms of [tex]\(x\)[/tex]:
- Number of members after [tex]\(x\)[/tex] increments: [tex]\(320 - 10x\)[/tex]
- Membership fee after [tex]\(x\)[/tex] increments: [tex]\(45 + 5x\)[/tex]
4. Write the revenue equation:
- Revenue, [tex]\(y\)[/tex], is given by the product of the number of members and the membership fee.
So,
[tex]\[
y = (\text{Number of members}) \times (\text{Membership fee})
\][/tex]
Substituting the expressions for the number of members and the membership fee in terms of [tex]\(x\)[/tex]:
[tex]\[
y = (320 - 10x)(45 + 5x)
\][/tex]
5. Expand and simplify the revenue equation:
- Expand the product:
[tex]\[
y = (320 - 10x)(45 + 5x)
\][/tex]
[tex]\[
y = 320 \times 45 + 320 \times 5x - 10x \times 45 - 10x \times 5x
\][/tex]
[tex]\[
y = 14400 + 1600x - 450x - 50x^2
\][/tex]
- Combine like terms:
[tex]\[
y = 14400 + (1600x - 450x) - 50x^2
\][/tex]
[tex]\[
y = 14400 + 1150x - 50x^2
\][/tex]
6. Final revenue equation:
[tex]\[
y = -50x^2 + 1150x + 14400
\][/tex]
Hence, the correct answer is:
[tex]\[
\boxed{A} \quad y = -50x^2 + 1150x + 14400
\][/tex]
