Question

Two companies modeled their profits for one yeor Which stotement describes the relationship between the profits, predicted by the models of the two companies?

Answer 1

The question is incomplete. The complete question is :

Two companies modeled their profits for one year.

• Company A used the function P(t)=1.8(1.4)^t to represent its monthly profit, P(t), in hundreds of dollars, after t months, where 0 < t ≤ 12.

• Company B used the data in the table to write a linear model to represent its monthly profits.

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 $P(t) = (1.8)(1.4)^t$ in order to represent the monthly profit of the company in hundreds of dollars after a time $t$.

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 :

$P(t)=mt+c$

Here, m = slope of line

          c = y-intercept

According to the data from the table , we see that the two points $(3.5)$ and $(4.10)$ lies on the line so that the slope of the line is represented by :

$\frac{y-y'}{x-x'}=\frac{10-5}{4-3}=5$

The point $(3.5)$ passes through the given line.

∴ $5=5(3)+c$

$5=15+c$

$c=5-15$

$c=-10$

Therefore, the function will be $P(t) = 5t-10$

So, at $t=4$,

the profit of the company A is $P(4)=(1.8)(1.4)^4$

                                                          = 6.91

the profit of the company A is $P(4)=5(4)-10$

                                                           = 20 - 10

                                                            = 10

Therefore,  $t=4$, the profit of the company B is more than the profit of company A.

Now at $t=12$,

Profit of company A is $P(12) =(1.8)(1.4)^{12}$

                                               = 102.05

Profit of company B is $P(12)=5(12)-10$

                                               $=66-10$

                                               = 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.