Question

From 2000 to 2020 a company’s net income in thousands of dollars can be modeled by f(t)=-0.05t^3+1.51t^2-11.12t+50.71. The company’s costs in thousands of dollars can be modeled by g(t)=0.04t^3+1.23t^2-8.57t+31.43 where t is the number of years since 2000
Find (f-g)(t) and explain what it represents

Answer 1

To find \((f-g)(t)\), we need to subtract the cost function \(g(t)\) from the net income function \(f(t)\).

We have:
- \(f(t) = -0.05t^3 + 1.51t^2 - 11.12t + 50.71\)
- \(g(t) = 0.04t^3 + 1.23t^2 - 8.57t + 31.43\)

Now, we can compute \((f-g)(t)\):

\[
(f-g)(t) = f(t) - g(t)
\]
\[
= \left(-0.05t^3 + 1.51t^2 - 11.12t + 50.71\right) - \left(0.04t^3 + 1.23t^2 - 8.57t + 31.43\right)
\]

Distributing the negative sign, we get:

\[
= -0.05t^3 + 1.51t^2 - 11.12t + 50.71 - 0.04t^3 - 1.23t^2 + 8.57t - 31.43
\]

Now, we can combine like terms:

1. **Cubic terms**:
\(-0.05t^3 - 0.04t^3 = -0.09t^3\)

2. **Quadratic terms**:
\(1.51t^2 - 1.23t^2 = 0.28t^2\)

3. **Linear terms**:
\(-11.12t + 8.57t = -2.55t\)

4. **Constant terms**:
\(50.71 - 31.43 = 19.28\)

Putting it all together, we have:

\[
(f-g)(t) = -0.09t^3 + 0.28t^2 - 2.55t + 19.28
\]

### Interpretation:

The function \((f-g)(t)\) represents the company's profit, measured in thousands of dollars. More specifically:

- If \((f-g)(t) > 0\), the company is making a profit in year \(t\) (since net income exceeds costs).
- If \((f-g)(t) < 0\), the company is incurring a loss in year \(t\) (since costs exceed net income).
- If \((f-g)(t) = 0\), the company breaks even, meaning its revenues are equal to its costs.

In summary, \((f-g)(t)\) provides insights into the company's profitability over the years from 2000 to 2020.