Question

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?

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

Answer 1

Answer: the correct answer is d) 1 month

Step-by-step explanation: To determine when the franchise will begin to show a profit, we need to find the value of t that makes the profitability function, P(t), greater than zero.

Given the function P(t) = t^4 - 9t^3 + 24t^2 - 20t, we can set it equal to zero and solve for t to find the points where the profitability is zero.

t^4 - 9t^3 + 24t^2 - 20t = 0

We can factor out a common factor of t:

t(t^3 - 9t^2 + 24t - 20) = 0

Now, we can solve the equation t^3 - 9t^2 + 24t - 20 = 0.

By trying different values for t, we find that when t = 1, the equation evaluates to zero. Therefore, the franchise will begin to show a profit 1 month after it opens.

I HOPE THIS HELPS!!!

Answer 2

Explanation:

To determine when the franchise will begin to show a profit, we need to find the values of t for which the profitability function P(t) is positive.

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

To find the values of t for which P(t) > 0, we need to solve the inequality:

t^4 - 9t^3 + 24t^2 - 20t > 0

We can factor out t to simplify the inequality:

t(t^3 - 9t^2 + 24t - 20) > 0

Now, we can see that t = 0 is a solution to the inequality. To find the other solutions, we can factor the cubic expression inside the parentheses:

t(t - 2)(t - 2)(t - 5) > 0

The solutions to this inequality are t = 0, t = 2, and t = 5.

Therefore, the franchise will begin to show a profit after 2 months.

So the correct answer is c) 2 months.