https://file.aiquickdraw.com/mathbot/file/1726909260058a444vpl3.jpg

The problem in your image is asking for the total accumulated profit when 40 items are produced and sold. The marginal profit is given as:

\[
P'(x) = 196.4e^{0.03463x}
\]

### Steps to Solve:

1. **Find the Total Accumulated Profit**:
   The total accumulated profit $ P(x) $ is found by integrating the marginal profit function $ P'(x) $. So, we need to compute:

   \[
   P(x) = \int P'(x) \, dx = \int 196.4e^{0.03463x} \, dx
   \]

2. **Perform the Integration**:
   The integral of an exponential function $ e^{kx} $ is:

   \[
   \int e^{kx} \, dx = \frac{1}{k} e^{kx}
   \]

   Applying this rule, we get:

   \[
   P(x) = \frac{196.4}{0.03463} e^{0.03463x} + C
   \]

3. **Evaluate the Constant**:
   Since we're looking for the total profit accumulated, we can assume that at $ x = 0 $, there is no profit, meaning $ P(0) = 0 $. This allows us to solve for $ C $:

   \[
   P(0) = \frac{196.4}{0.03463} e^{0} + C = 0
   \]
   \[
   C = -\frac{196.4}{0.03463}
   \]

   So the total accumulated profit function becomes:

   \[
   P(x) = \frac{196.4}{0.03463} e^{0.03463x} - \frac{196.4}{0.03463}
   \]

4. **Calculate the Accumulated Profit for 40 Items**:
   Now substitute $ x = 40 $ into the profit function:

   \[
   P(40) = \frac{196.4}{0.03463} e^{0.03463 \times 40} - \frac{196.4}{0.03463}
   \]

   Let's calculate the result.The total accumulated profit when 40 items are produced and sold is **$16,989.34**.

Would you like more details on any part of this solution, or have any questions?

Here are 5 related questions to consider:
1. How would the accumulated profit change if the growth rate $ 0.03463 $ were different?
2. How can you compute the accumulated profit if the marginal profit function was not exponential?
3. What is the interpretation of the constant $ C $ in the context of the total profit function?
4. How does the marginal profit function impact business decisions when pricing products?
5. What would happen to the profit if production increases from 40 to 50 items?

**Tip:** In problems involving exponential growth or decay, integration often simplifies calculations involving cumulative totals.

A company finds that the marginal profit, in dollars per item, from producing and selling x items is given by P'(x) = 196.4e^{0.03463x}. Find the total accumulated profit when 40 items are produced and sold.

Learn how to calculate the total accumulated profit when producing and selling 40 items using the marginal profit function P'(x) = 196.4e^{0.03463x}. This problem covers integral calculus, exponential functions, and their application in business. Suitable for college-level calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation