The question is incomplete. The complete question is :
Two companies modeled their profits for one year.
Which statement describes the relationship between the profits, predicted be the models, of the two companies?
Solution :
For one year, the two companies A and B modeled their profits.
It is given that :
Company A uses function [tex]$P(t) = (1.8)(1.4)^t$[/tex] in order to represent the monthly profit of the company in hundreds of dollars after a time [tex]t[/tex].
But the company B uses the data in the table in order to write the linear model to represent their monthly income.
We know the linear function is given by :
[tex]$P(t)=mt+c$[/tex]
Here, m = slope of line
c = y-intercept
According to the data from the table , we see that the two points [tex](3.5)[/tex] and [tex](4.10)[/tex] lies on the line so that the slope of the line is represented by :
[tex]$\frac{y-y'}{x-x'}=\frac{10-5}{4-3}=5$[/tex]
The point [tex](3.5)[/tex] passes through the given line.
∴ [tex]$5=5(3)+c$[/tex]
[tex]$5=15+c$[/tex]
[tex]$c=5-15$[/tex]
[tex]$c=-10$[/tex]
Therefore, the function will be [tex]$P(t) = 5t-10$[/tex]
So, at [tex]t=4[/tex],
the profit of the company A is [tex]$P(4)=(1.8)(1.4)^4$[/tex]
= 6.91
the profit of the company A is [tex]$P(4)=5(4)-10$[/tex]
= 20 - 10
= 10
Therefore, [tex]t=4[/tex], the profit of the company B is more than the profit of company A.
Now at [tex]t=12[/tex],
Profit of company A is [tex]$P(12) =(1.8)(1.4)^{12}$[/tex]
= 102.05
Profit of company B is [tex]P(12)=5(12)-10[/tex]
[tex]$=66-10$[/tex]
= 56
Therefore, the profit of company A is more that that of company B at the end of the year one.
Thus, company B had a greater profit for the fourth month and ended the year with the greater monthly profits than company A.
Option (B) is the correct answer.
