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.