The profitability of the restaurant is when the function gives a value greater than 0. The franchise does not make a profit after 5, 3, 2 or 1 month of opening.
Given that:
[tex]P(t) = t^4 - 9t^3 + 24t^2 - 20t[/tex]
To calculate the profit, we simply substitute the values of t in the above equation.
(a) 5 months
This means [tex]t =5[/tex]
So, we have:
[tex]P(5) = 5^4 - 9\times 5^3 + 24 \times 5^2 - 20 \times 5[/tex]
[tex]P(5) = 0[/tex]
(b) 3 months
This means that [tex]t=3[/tex]
So, we have:
[tex]P(3) = 3^4 - 9 \times 3^3 + 24 \times 3^2 - 20 \times 3[/tex]
[tex]P(3) = -6[/tex]
(c) 2 months
This means that [tex]t = 2[/tex]
So, we have:
[tex]P(2) = 2^4 - 9 \times 2^3 + 24 \times 2^2 - 20 \times 2[/tex]
[tex]P(2) = 0[/tex]
(d) 1 month
This means that [tex]t = 1[/tex]
So, we have:
[tex]P(1) = 1^4 - 9 \times 1^3 + 24 \times 1^2 - 20 \times 1[/tex]
[tex]P(1) = -4[/tex]
For the restaurant to make a profit, the equation has to give a positive value. Since none of the values of t gives a positive value for P(t).
Then, we can say the franchise does not make a profit after 5, 3, 2 or 1 month of opening.
Read more about functions at:
https://brainly.com/question/24314573
