The net profit of a company was $721 million in 2010 and $869 million in
2011.
a) Using only this information, write a linear equation that models the net
profit P in terms of the year t. (Let t-O represent 2010.)
P-
b) Use the model to predict the profit for 2012. (Don't forget to change your
answer to billion. One billion is equal to one thousand million. You can find the
number of billions by dividing the number of millions by 1000.)
$
billion

To write a linear equation modeling the net profit P in terms of the year t, we can use the given data points: (t₁, P₁) = (0, 721) for the year 2010 and (t₂, P₂) = (1, 869) for the year 2011.

First, find the slope (m):

[tex]m = \dfrac{{P_2 - P_1}}{{t_2 - t_1}}\\\\\\m = \dfrac{{869 - 721}}{{1 - 0}}\\\\\\m = \dfrac{{148}}{{1}} \\\\\\m= 148[/tex]

Now that we have the slope, we can use the point-slope form of a linear equation to write the equation:

[tex]P - P_1 = m(t - t_1)\\\\\\P - 721 = 148(t - 0)\\\\\\P - 721 = 148t\\\\\\P = 148t + 721[/tex]

So, the linear equation that models the net profit P in terms of the year t, where t = 0 represents 2010, is

[tex]\Large\boxed{\boxed{P = 148t + 721}}[/tex]

[tex]\dotfill[/tex]

Part b)

Since t = 0 represents 2010, the year 2012 would be represented by t = 2. Therefore, to predict the profit for 2012, substitute t = 2 into the equation:

[tex]P = 148(2) + 721 \\\\\\P = 296 + 721 \\\\\\ P = 1017\; \text{million}[/tex]

To convert this to billions, divide by 1000 (since 1 billion is equal to 1000 million):

[tex]P = \dfrac{{1017}}{{1000}} = 1.017 \text{ billion}[/tex]

So, the predicted profit for 2012 is:

[tex]\Large\boxed{\boxed{\$1.017 \; \rm billion}}[/tex]