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The problem provides the derivative \( S'(t) \) for the rate of sales decline of a computer over time \( t \), measured in months. Here is a breakdown of the information given:

1. **Given derivative (rate of sales)**: 
   \[
   S'(t) = -20t^2 - 60
   \]

2. **Condition for sales stopping**:
   The company will stop manufacturing this computer when the monthly sales reach 800 units.

3. **Initial sales (when \( t = 0 \))**:
   The sales start at 1,400 computers.

4. **Goal**:
   To find the function \( S(t) \), which represents the number of computers sold each month as a function of time \( t \), and then to solve for \( t \) when \( S(t) = 800 \).

### Solution Steps

1. **Integrate \( S'(t) \) to find \( S(t) \):**
   Since \( S'(t) = -20t^2 - 60 \), we integrate this to get \( S(t) \).

   \[
   S(t) = \int (-20t^2 - 60) \, dt
   \]

2. **Perform the integration:**
   \[
   S(t) = -\frac{20}{3}t^3 - 60t + C
   \]
   where \( C \) is the integration constant.

3. **Determine the constant \( C \) using the initial condition \( S(0) = 1400 \):**
   Substitute \( t = 0 \) and \( S(0) = 1400 \):
   \[
   S(0) = -\frac{20}{3} \cdot 0^3 - 60 \cdot 0 + C = 1400
   \]
   Thus, \( C = 1400 \).

4. **Construct the sales function \( S(t) \):**
   \[
   S(t) = -\frac{20}{3}t^3 - 60t + 1400
   \]

5. **Set up the equation to find \( t \) when \( S(t) = 800 \):**
   \[
   -\frac{20}{3}t^3 - 60t + 1400 = 800
   \]

6. **Simplify the equation:**
   \[
   -\frac{20}{3}t^3 - 60t + 600 = 0
   \]

   You would use a graphing calculator to approximate the value of \( t \) for which this equation holds.

### Solution Check with Graphing Calculator

Using a graphing tool or calculator, plot the equation:
\[
S(t) = -\frac{20}{3}t^3 - 60t + 1400
\]
and find the value of \( t \) when \( S(t) = 800 \).

Would you like more details on this process or have any questions?

---

### Follow-up Questions:

1. How do you perform the integration step-by-step to obtain \( S(t) \)?
2. What methods can be used to solve for \( t \) in the equation \( S(t) = 800 \)?
3. How do initial conditions affect the integration constant in \( S(t) \)?
4. What is the significance of \( S'(t) \) in terms of real-world interpretation?
5. How can a graphing calculator help in finding roots of polynomial equations like this?

### Tip:

When integrating a function to find the original function, remember to always add the constant of integration \( C \) and use any initial conditions to determine its value.

Monthly sales of a particular computer are expected to decline at the following rate of S'(t) computers per month, where t is time in months and S(t) is the number of computers sold each month:

S'(t) = -20t^2 - 60

The company plans to stop manufacturing this computer when monthly sales reach 800 computers. If monthly sales now (t = 0) are 1,400 computers, find S(t). Use a graphing calculator to approximate the solution of the equation S(t) = 800.

Calculate the monthly sales function S(t) for a computer product declining at a rate of S'(t) = -20t^2 - 60. Use initial conditions and integration to determine when sales reach 800 units, starting from 1,400. This calculus problem is ideal for students in Grades 11-12, covering integration and polynomial functions.

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