https://file.aiquickdraw.com/mathbot/file/1730137690371fbgjarzk.jpg

To solve this problem, we need to understand how to model the company's profit over time based on the given revenue and costs.

### Given Information:
1. **Revenue for 2014**: $75,112.
2. **Revenue Growth Rate**: 4% annually.
3. **Cost for 2014**: $31,405.
4. **Cost Growth Rate**: $3,000 increase each year.

Let $ x $ represent the number of years after 2014.

### Step 1: Model the Revenue Function
Since the revenue grows by 4% each year, the revenue function can be expressed as:
\[
\text{Revenue} = 75,112 \times (1.04)^x
\]

### Step 2: Model the Cost Function
The cost starts at $31,405 in 2014 and increases by $3,000 each year. Thus, the cost function can be expressed as:
\[
\text{Cost} = 31,405 + 3,000x
\]

### Step 3: Define the Profit Function
Profit is calculated as the revenue minus the cost. Therefore:
\[
f(x) = \text{Revenue} - \text{Cost}
\]
Substitute the revenue and cost expressions:
\[
f(x) = 75,112 \times (1.04)^x - (31,405 + 3,000x)
\]

### Step 4: Simplify and Match to Options
The profit function that best represents this setup is:
\[
f(x) = 75,112 \times (1.04)^x - 3,000x - 31,405
\]

### Answer:
The correct choice is:
\[
\boxed{\text{C: } f(x) = 75,112(1.04)^x - 3,000x - 31,405}
\]

Would you like further details on any of these steps?

---

1. What if the revenue growth rate was different, say 5%?
2. How would the function change if the cost increased by a percentage rather than a fixed amount?
3. What would be the profit in 2020 using this function?
4. How does exponential growth differ from linear growth?
5. Can we use this function to calculate losses as well?

**Tip:** When calculating changes over time, always note if growth is exponential (percentage-based) or linear (fixed amount) for accuracy in modeling.

A small company sells a product that earned $75,112 in revenue for 2014, and it expects to earn 4% more in revenue each year. The company’s costs for 2014 were $31,405, and they are expected to increase by $3,000 each year. If x represents the number of years after 2014, which function best represents the profit for the company?

Calculate the profit function for a company with an initial revenue of $75,112 that grows by 4% annually and costs starting at $31,405, increasing by $3,000 each year. This problem involves exponential growth, linear growth, and algebraic modeling. Ideal for students in Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation