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The image you provided includes a mathematical problem asking to find a model $ R(t) $ for the revenue of "Mom & Pop's Diner" over time, based on the rate of change of revenue and an initial condition for the revenue in 2002.

Here’s the breakdown of the problem:

### Given Information:
- The rate of change of the revenue $ r(t) $ is modeled by:
  \[
  r(t) = 28.512t - 58.663
  \]
  where $ t $ represents the number of years after 2000, and $ r(t) $ is in thousand dollars per year.
  
- Initial revenue in 2002: $ R(2) = 640 $ thousand dollars.

### Required:
- Find the function $ R(t) $, which represents the revenue of the diner over time, based on the given rate of change and the initial condition.
  
### Solution Outline:
1. **Find $ R(t) $ from $ r(t) $**: 
   $ r(t) $ represents the derivative of $ R(t) $, i.e., $ r(t) = R'(t) $.
   
   To find $ R(t) $, integrate $ r(t) $.
   
   \[
   R(t) = \int (28.512t - 58.663) \, dt
   \]
   
2. **Integrate**: 
   \[
   R(t) = 28.512 \cdot \frac{t^2}{2} - 58.663t + C
   \]
   Simplifying:
   \[
   R(t) = 14.256t^2 - 58.663t + C
   \]
   
3. **Use the initial condition $ R(2) = 640 $** to solve for $ C $:
   \[
   R(2) = 14.256(2)^2 - 58.663(2) + C = 640
   \]
   \[
   R(2) = 14.256 \cdot 4 - 117.326 + C = 640
   \]
   \[
   57.024 - 117.326 + C = 640
   \]
   \[
   -60.302 + C = 640
   \]
   \[
   C = 700.302
   \]

4. **Final Model**:
   \[
   R(t) = 14.256t^2 - 58.663t + 700.302
   \]
   
This is the model that represents the revenue $ R(t) $ for the diner for $ t $ years after 2000.

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Would you like further clarification or details on any part of the solution?

Here are five follow-up questions to expand your understanding:
1. How would the model change if the initial revenue in 2002 was different?
2. Can this model be used to predict the revenue beyond 2015, and what considerations should be taken?
3. How does the rate of change $ r(t) $ affect the shape of the revenue curve?
4. How do you interpret the constant $ C $ in terms of the revenue model?
5. How would the integration process change if $ r(t) $ were a more complex function?

**Tip:** When solving differential equations, always remember to use any given initial conditions to find the integration constant.

Given that Mom & Pop's revenue was $640 thousand in 2002, find a model R(t) which can be used to calculate Mom & Pop's revenue t years after 2000.

Learn how to calculate the revenue of Mom & Pop's Diner over time using integration. This detailed solution includes a step-by-step breakdown of the rate of change formula and uses an initial revenue value to find the function R(t). Suitable for Grades 10-12.

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