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10. An accountant finds that the gross income, in thousands of dollars, of a small business can be modeled by the polynomial [tex]-0.3 t^2 + 8t + 198[/tex], where [tex]t[/tex] is the number of years after 2010. The yearly expenses of the business, in thousands of dollars, can be modeled by the polynomial [tex]-0.2 t^2 + 2t + 131[/tex].

a. Find a polynomial that predicts the net profit of the business after [tex]t[/tex] years.

b. Assuming that the models continue to hold, how much net profit can the business expect to make in the year 2016?



Answer :

GinnyAnswer
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.

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