A tourist agency’s profit P( in dollars) can be modelled by the function P(x)= -25x^2+1000x-3000, where x is the number of tourists. Determine how many tourists are needed for the agency’s profit to be at least $5000.

We are given the profit function:

\[
P(x) = -25x^2 + 1000x - 3000
\]

We want to find how many tourists $ x $ are needed for the agency's profit to be at least $5000. This means solving the inequality:

\[
P(x) \geq 5000
\]

Substitute 5000 for $ P(x) $:

\[
-25x^2 + 1000x - 3000 \geq 5000
\]

Now, simplify this inequality:

\[
-25x^2 + 1000x - 3000 - 5000 \geq 0
\]

\[
-25x^2 + 1000x - 8000 \geq 0
\]

To simplify further, divide the entire inequality by $-25$ (note that dividing by a negative flips the inequality sign):

\[
x^2 - 40x + 320 \leq 0
\]

We now solve the quadratic inequality:

1. Solve the quadratic equation $ x^2 - 40x + 320 = 0 $ using the quadratic formula:

\[
x = \frac{-(-40) \pm \sqrt{(-40)^2 - 4(1)(320)}}{2(1)}
\]

\[
x = \frac{40 \pm \sqrt{1600 - 1280}}{2}
\]

\[
x = \frac{40 \pm \sqrt{320}}{2}
\]

\[
x = \frac{40 \pm 17.89}{2}
\]

The two solutions are:

\[
x_1 = \frac{40 + 17.89}{2} = 28.945 \quad \text{(approximately 29)}
\]
\[
x_2 = \frac{40 - 17.89}{2} = 11.055 \quad \text{(approximately 11)}
\]

Thus, the solutions to the quadratic equation are approximately $ x = 11 $ and $ x = 29 $.

2. Since we have $ x^2 - 40x + 320 \leq 0 $, the values of $ x $ between 11 and 29 satisfy this inequality.

### Conclusion:
The number of tourists needed for the agency's profit to be at least $5000 is between **11 and 29 tourists**.

Do you have any questions or would you like further details?

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Here are 5 related questions for further exploration:
1. What is the maximum profit the agency can achieve?
2. How many tourists are needed for the agency to break even (profit = 0)?
3. What happens to the profit if the number of tourists exceeds 29?
4. How would the profit function change if costs were increased by $2000?
5. How does the shape of the profit function graph help determine optimal tourist numbers?

**Tip:** When solving quadratic inequalities, graphing the function can help visualize which intervals satisfy the inequality.

A tourist agency’s profit P (in dollars) can be modeled by the function P(x) = -25x^2 + 1000x - 3000, where x is the number of tourists. Determine how many tourists are needed for the agency’s profit to be at least $5000.

Learn how to determine the number of tourists needed for a tourist agency’s profit to be at least $5000. This problem explores quadratic functions, inequalities, and profit maximization, offering a detailed step-by-step solution. Suitable for students in Grades 9-11.

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