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
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.
To determine when the franchise will begin to show a profit, we need to find the value of t when the profitability function P(t) is greater than 0.
P(t) = t^4 - 9t^3 + 24t^2 - 20t
We need to solve the equation:
t^4 - 9t^3 + 24t^2 - 20t > 0
To solve this inequality, we can factor out a common factor of t:
t(t^3 - 9t^2 + 24t - 20) > 0
Now, let's factor the quartic equation in the parentheses:
t(t - 2)(t - 2)(t - 5) > 0
To determine the critical points, we set each factor equal to zero and solve for t:
t = 0, t - 2 = 0, t - 2 = 0, t - 5 = 0
From these equations, we find that t = 0, 2, and 5.
Now, let's analyze the signs of the factors:
When t < 0, all the factors are negative.
When 0 < t < 2, all the factors are positive.
When 2 < t < 5, the factors t - 2 and t - 5 are negative, and t is positive.
When t > 5, all the factors are positive.
We are interested in values of t for which the expression t(t - 2)(t - 2)(t - 5) is positive.
From the analysis of the signs, it is clear that t must be between 2 and 5 to satisfy the inequality.
Therefore, the franchise will begin to show a profit between 2 and 5 months after it opens.
Based on the options provided, the correct answer is:
c) 2 months
To determine when the restaurant franchise will begin to show a profit, we need to find the value of t when the profitability function P(t) is greater than zero.
Step 1: Set up the inequality:
P(t) > 0
Step 2: Set up the function:
P(t) = t^4 - 9t^3 + 24t^2 - 20t
Step 3: Solve the inequality:
t^4 - 9t^3 + 24t^2 - 20t > 0
To solve this inequality, we can factor out a common factor t:
t(t^3 - 9t^2 + 24t - 20) > 0
Next, we can find the roots of the equation t^3 - 9t^2 + 24t - 20 = 0. By using synthetic division or a graphing calculator, we find that the roots are t = 1, t = 2, and t ≈ 5.39.
Step 4: Use the roots to determine the intervals:
Now that we know the roots of the equation, we can create intervals on the number line based on these roots. The inequality will be true for the intervals where the sign of the expression changes.
The intervals are:
Interval 1: (-∞, 1)
Interval 2: (1, 2)
Interval 3: (2, 5.39)
Interval 4: (5.39, ∞)
Step 5: Determine when the expression is positive or negative on each interval:
To determine the sign of the expression on each interval, we can test a point within each interval.
Interval 1: Choose t = 0:
Substitute t = 0 into the inequality:
0(t^3 - 9t^2 + 24t - 20) > 0
-20 < 0
Since the inequality is false, the expression is negative on this interval.
Interval 2: Choose t = 1.5 (a value between 1 and 2):
Substitute t = 1.5 into the inequality:
1.5(1.5^3 - 9(1.5)^2 + 24(1.5) - 20) > 0
3(1.5^3) - 27(1.5^2) + 36(1.5) - 30 > 0
7.875 - 60.75 + 54 - 30 > 0
-27.875 < 0
Since the inequality is false, the expression is negative on this interval.
Interval 3: Choose t = 3 (a value between 2 and 5.39):
Substitute t = 3 into the inequality:
3(3^3 - 9(3)^2 + 24(3) - 20) > 0
3(3^3) - 27(3^2) + 36(3) - 30 > 0
81 - 243 + 108 - 30 > 0
-84 < 0
Since the inequality is false, the expression is negative on this interval.
Interval 4: Choose t = 6:
Substitute t = 6 into the inequality:
6(6^3 - 9(6)^2 + 24(6) - 20) > 0
6(6^3) - 27(6^2) + 36(6) - 30 > 0
6(216) - 27(36) + 36(6) - 30 > 0
1296 - 972 + 216 - 30 > 0
510 > 0
Since the inequality is true, the expression is positive on this interval.
Step 6: Determine the answer based on the intervals:
The expression is positive on the interval (5.39, ∞). Since 5 months is within this interval, the answer is 5 months.
Therefore, the correct option is:
a) 5 months