The profitability, P, of a popular restaurant franchise can be modeled by the function P (t) = t4 − 9t3 + 24t2 − 20t, 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?

Question

a) 5 months
b) 3 months
c) 2 months
d) 1 month

Answer 1

To find when the franchise will begin to show a profit, we need to find the value of t when P(t) is positive.

P(t) = t^4 - 9t^3 + 24t^2 - 20t

Let's factor out t from each term:
P(t) = t(t^3 - 9t^2 + 24t - 20)

Now, we can use the factor theorem to find if there are any integer solutions for t. Checking the factors of 20, we find that t = 1 is a root, since P(1) = 0. Therefore, (t - 1) is a factor of t^3 - 9t^2 + 24t - 20.

Dividing t^3 - 9t^2 + 24t - 20 by (t - 1), we get:
t^3 - 9t^2 + 24t - 20 = (t - 1)(t^2 - 8t + 20)

Now, let's solve the quadratic equation t^2 - 8t + 20 = 0. We can use the quadratic formula to find the roots:
t = (-b ± √(b^2 - 4ac)) / (2a)
t = (-(-8) ± √((-8)^2 - 4(1)(20))) / (2(1))
t = (8 ± √(64 - 80)) / 2
t = (8 ± √(-16)) / 2

Since we have a negative number inside the square root, there are no real solutions for t. This means that the quadratic equation does not intersect the x-axis, which means P(t) is always negative for all values of t.

Therefore, the franchise will not begin to show a profit. The answer is none of the given options.