Question

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?

Answer 1

we know the equation for company A is P(t) = 1.8(1.4)ᵗ.

and what we know about company B is that their equation is linear, and we also have 3 months in a table of values, and we can simply use two points from there to get the equation of company B, let's do so using (3,5) and (7, 25),

$\bf \begin{array}{ccccccccc} &&x_1&&y_1&&x_2&&y_2\\ % (a,b) &&(~ 3 &,& 5~) % (c,d) &&(~ 7 &,& 25~) \end{array} \\\\\\ % slope = m slope = m\implies \cfrac{\stackrel{rise}{ y_2- y_1}}{\stackrel{run}{ x_2- x_1}}\implies \cfrac{25-5}{7-3}\implies \cfrac{20}{4}\implies 5 \\\\\\ % point-slope intercept \stackrel{\textit{point-slope form}}{y- y_1= m(x- x_1)}\implies y-5=5(x-3) \\\\\\ y-5=5x-15\implies y=5x-10\implies p(t)=5t-10$

now, the end of the year will be at the twelfth month, namely t = 12, and the four month well we know that t = 4.

how did each one fare?

$\bf \stackrel{\textit{at the 4th month}}{\stackrel{\textit{company A}}{P(4)=6.91488}\qquad \qquad \stackrel{\textit{company B}}{p(4)=10}} \\\\\\ \stackrel{\textit{at the end of the year}}{\stackrel{\textit{company A}}{P(12)\approx 102.04904}\qquad \qquad \stackrel{\textit{company B}}{p(12)=70}}$

so, notice, on the fourth month B did better than A, but at the end of the year, A ended up better off than B.

Answer 2

Company B follows linear profit function while company A follows exponential profit function. Company B has more profit after 4 months while company A has more monthly profit after year-end. Option A is correct.

Given information:

Company A earns monthly profit based on the profit function, $P(t)=1.8(1.4)^t$ where t is the number of months.

Company B earns profit based on a linear function. The data in the below table shows the profit after each month:

Month 3 4 7

Profit 5 10 25

Now, the profit earned by company A after 4 months will be,

$P(4)=1.8(1.4)^4\\P(4)=1.8\times 3.8416\\P(4)=6.915$

Also, the profit earned after a year will be,

$P(12)=1.8(1.4)^{12}\\P(4)=102.04$

Now, the linear function which represents the profit of company B is,

$P-5=\dfrac{10-5}{4-3}(t-3)\\P-5=5t-15\\P=5t-10$

So, the profit earned after 4 months and 1 year (by company B) will be,

$P(4)=5\times 4-10=10\\P(12)=5\times 12-10=50$

Now, comparing the profit of companies A and B:

After 4 months, company B has more monthly profit which is 10 hundred dollars or 1 thousand dollars. After the year-end, company A earns more monthly profit as it is 102.04 hundred dollars. This is because company A follows an exponential function to predict the profit while company B uses a linear profit function.

Therefore, option A should be correct.

For more details, refer to the link:

https://brainly.com/question/14628115