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.