Question

An online retail company determined that their cost for each day, x, can be determined by the function C(x) = 2x2 – 20x + 80 and their revenue by the function R(x) = –x2 + 5x + 58. How many days will it take for the company to earn a profit?

32
30
5
1

Answer 1

To find the number of days it will take for the company to earn a profit, we need to determine when the revenue exceeds the cost. We solve the inequality -x^2 + 5x + 58 > 2x^2 - 20x + 80 and find that x > 2. Thus, it will take at least 3 days for the company to earn a profit.

Step-by-step explanation:

To determine how many days it will take for the company to earn a profit, we need to find the point where the revenue is greater than the cost. In other words, we need to find the x-value at which R(x) > C(x).

Given the functions C(x) = 2x^2 - 20x + 80 and R(x) = -x^2 + 5x + 58, we need to solve the inequality -x^2 + 5x + 58 > 2x^2 - 20x + 80.

Simplifying the inequality, we get 3x^2 - 15x + 22 > 0. Factoring the quadratic, we have (x - 2)(3x - 11) > 0. Solving for x, we find two possible solutions: x > 2 or x < 11/3.

Since it is not possible to have a negative number of days, we choose x > 2 as the solution. Therefore, it will take at least 3 days for the company to earn a profit.

Answer 2

It will take the company at least **3 days** to earn a profit.

The company will earn a profit when its revenue (R(x)) exceeds its cost (C(x)). Therefore, we need to find the number of days (x) when the inequality R(x) > C(x) holds true.

Here's how to solve it:

1. **Set up the inequality:**

R(x) > C(x)

2. **Substitute the given functions:**

(-x^2 + 5x + 58) > (2x^2 - 20x + 80)

3. **Combine like terms and simplify:**

3x^2 - 15x + 22 > 0

4. **Factor the inequality:**

(x - 2)(3x - 11) > 0

5. **Analyze the signs:**

Since the inequality is greater than 0, either both factors must be positive or both must be negative. We can analyze the signs of each factor for different values of x:

* x < 2: Both factors are negative.

* 2 < x < 11/3: Only the first factor is positive. (Not a solution as R(x) will be negative)

* x > 11/3: Both factors are positive.

6. **Conclusion:**

Therefore, the company will earn a profit only when x > 11/3. Since we are looking for the number of days, we round the value up to the nearest whole day, which is **3 days**.

Therefore, it will take the company at least **3 days** to earn a profit.