Question

The profitability, P, of a popular restaurant franchise can be modeled by the function P (t) = t4 – 10t3 + 28t2 – 24t, where t is the number of months since the restaurant opened. How many months after the franchise opens will it begin to show a profit?

A. 0 months
B. 2 months
C. 4 months
D. 6 months

Answer 1

In this question, the profit of the restaurant after t months is given by a polynomial function. To find when it begins to show a profit, we find the numerical values of the function for t, and it shows a profit when $t > 0$

Profit after t months:

$P(t) = t^4 - 10t^3 + 28t^2 - 24t$

0 months:

This is P(0). So

$P(0) = 0^4 - 10*0^3 + 28*0^2 - 24*0 = 0$

1 month:

This is P(1). So

$P(1) = 1^4 - 10*1^3 + 28*1^2 - 24*1 = -5$

2 months:

This is P(2). So

$P(2) = 2^4 - 10*2^3 + 28*2^2 - 24*2 = 0$

3 months:

This is P(3). So

$P(3) = 3^4 - 10*3^3 + 28*3^2 - 24*3 = -9$

4 months:

This is P(4). So

$P(4) = 4^4 - 10*4^3 + 28*4^2 - 24*4 = -32$

5 months:

This is P(5). So

$P(5) = 5^4 - 10*5^3 + 28*5^2 - 24*5 = -45$

6 months:

This is P(6). So

$P(6) = 6^4 - 10*6^3 + 28*6^2 - 24*6 = 0$

7 months:

This is `P(7). So

$P(7) = 7^4 - 10*7^3 + 28*7^2 - 24*7 = 175$

After 7 months it shows profit, so it starts showing profit on the 6th month, and thus, the correct answer is given by option D.

For another example of a function involving numeric value, you can check https://brainly.com/question/24231879