Sure, let's solve the given problem step-by-step:
### a. Finding the polynomial that predicts the net profit of the business
The net profit of a business is calculated as the difference between the gross income and the expenses. Given the polynomials for gross income and expenses:
- Gross Income: [tex]\(G(t) = -0.3 t^2 + 8 t + 198\)[/tex]
- Expenses: [tex]\(E(t) = -0.2 t^2 + 2 t + 131\)[/tex]
The net profit polynomial [tex]\(P(t)\)[/tex] is:
[tex]\[P(t) = G(t) - E(t)\][/tex]
Substitute the given polynomials:
[tex]\[P(t) = (-0.3 t^2 + 8 t + 198) - (-0.2 t^2 + 2 t + 131)\][/tex]
Distribute the negative sign:
[tex]\[P(t) = -0.3 t^2 + 8 t + 198 + 0.2 t^2 - 2 t - 131\][/tex]
Now, combine like terms:
[tex]\[
\begin{align*}
P(t) &= (-0.3 t^2 + 0.2 t^2) + (8 t - 2 t) + (198 - 131) \\
P(t) &= -0.1 t^2 + 6 t + 67
\end{align*}
\][/tex]
So, the polynomial that predicts the net profit [tex]\(P(t)\)[/tex] of the business after [tex]\(t\)[/tex] years is:
[tex]\[P(t) = -0.1 t^2 + 6 t + 67\][/tex]
### b. Calculating the net profit expected in the year 2016
To find the net profit for the year 2016, we need to determine the value of [tex]\(t\)[/tex] for 2016. Since [tex]\(t\)[/tex] is the number of years after 2010:
[tex]\[2016 - 2010 = 6\][/tex]
Thus, [tex]\(t = 6\)[/tex].
Substitute [tex]\(t = 6\)[/tex] into the polynomial [tex]\(P(t)\)[/tex]:
[tex]\[
\begin{align*}
P(6) &= -0.1 (6)^2 + 6 (6) + 67 \\
P(6) &= -0.1 (36) + 36 + 67 \\
P(6) &= -3.6 + 36 + 67 \\
P(6) &= 99.4
\end{align*}
\][/tex]
So, the net profit the business can expect to make in the year 2016 is [tex]\(99.4\)[/tex] thousand dollars.
