The profit of a football club after a takeover is modeled by p(t)=tE3-14tE2+20t+120, where t is the number of years after the takeover. In which years was the club making a loss?
you want
t^3 - 14t^2 + 20t + 12 < 0
This has no rational roots, but it does have zeroes at
t = -0.45, 2.16, 12.29
From what you know about the shape of cubic curves, you know that
p(t) < 0 for 2.16 < t < 12.29
so the team took a loss in years 3-12
Well, it seems like this football club's financial situation is a bit of a rollercoaster, doesn't it? To find the years when the club was making a loss, we need to look for the values of t where the profit, p(t), is negative.
Let me just put on my mathematical clown shoes and solve this equation for you:
p(t) = t^3 - 14t^2 + 20t + 120
Now, we want to find the values of t when p(t) is less than 0. In other words, we want to find the years when the profit is in the red.
Unfortunately, I can't calculate the exact values for you without any numerical data for E. However, I can tell you that you'll need to solve the equation p(t) < 0.
It might be worth putting on your detective hat, gathering some numerical data, and plugging it into the equation to see which years the club was losing money.
To find the years when the club was making a loss, we need to identify the values of t for which the profit, p(t), is negative.
The given profit function is: p(t) = t^3 - 14t^2 + 20t + 120
To find the years when the club was making a loss, we set p(t) less than zero (negative):
t^3 - 14t^2 + 20t + 120 < 0
To solve this inequality, we can use various methods such as factoring, graphing, or using a sign chart. Let's solve it by using a sign chart.
1. First, let's find the values of t for which p(t) is equal to zero (the "zeros" or "roots" of the function). These values give us the boundaries for the intervals of the sign chart.
Setting p(t) = 0:
t^3 - 14t^2 + 20t + 120 = 0
We can solve this equation using synthetic division or factoring, or by using a graphing calculator to find the equal roots.
By solving the equation, we find that one of the roots is t = -5.
2. Now, consider three intervals in the sign chart: t < -5, -5 < t < ?, and ? < t.
Let's test a value from each interval to determine the sign of p(t) in that interval:
For t = -6:
p(-6) = (-6)^3 - 14(-6)^2 + 20(-6) + 120
= -216 - 504 + (-120) + 120
= -720
Since -720 < 0, p(t) is negative in the interval t < -5.
For t = -4:
p(-4) = (-4)^3 - 14(-4)^2 + 20(-4) + 120
= -64 - 224 + (-80) + 120
= -248
Since -248 < 0, p(t) is negative in the interval -5 < t < ?.
For t = 0:
p(0) = (0)^3 - 14(0)^2 + 20(0) + 120
= 120
Since 120 > 0, p(t) is positive in the interval ? < t.
Therefore, the club is making a loss during the years when t < -5.
To find the years during which the football club was making a loss, we need to determine the values of t for which the profit function p(t) is negative.
The profit function is given by:
p(t) = t^3 - 14t^2 + 20t + 120
To find the years of losses, we need to find the values of t that make p(t) negative, i.e., p(t) < 0.
The first step is to set up the inequality:
p(t) < 0
Now, substitute the expression of p(t) into the inequality:
t^3 - 14t^2 + 20t + 120 < 0
Next, we need to solve this inequality. You can use either graphical methods or algebraic methods (factoring, long division, synthetic division, etc.) to find the values of t that satisfy the inequality.
For simplicity, let's use algebraic methods to solve the inequality. Start by factoring out the common factor, if any.
p(t) = t^3 - 14t^2 + 20t + 120
= t(t^2 - 14t + 20) + 120
The quadratic expression inside the parentheses, t^2 - 14t + 20, can be factored further. We need to find two values that multiply to give 20 and add up to -14. The factors are 10 and 2.
Therefore, t^3 - 14t^2 + 20t + 120 = t(t - 10)(t - 2) + 120
Now, rewrite the inequality with factored form:
t(t - 10)(t - 2) + 120 < 0
The next step is to determine the intervals on the number line for which the inequality is true. We should consider the critical values of t where the expression changes sign. These critical values are obtained by setting each factor equal to zero:
t - 10 = 0 => t = 10
t - 2 = 0 => t = 2
Now, plot these critical values on the number line and test the regions between them to determine whether p(t) is positive or negative in each interval.
The number line:
-∞.........2........10.........∞
To test the intervals, we can use any value within each region and substitute it into the inequality to verify if it is satisfied.
Let's consider the interval (-∞, 2). To check if p(t) is negative in this interval, we can pick a value, let's say t = 0, and substitute it into the inequality:
p(0) = 0(0 - 10)(0 - 2) + 120
= 0 + 120
= 120
Since p(0) = 120 is positive, the inequality is not satisfied, and the club is not making a loss during this period.
Next, let's consider the interval (2, 10). Similarly, we can choose a value, let's say t = 5, and substitute it into the inequality:
p(5) = 5(5 - 10)(5 - 2) + 120
= -5(-5)(3) + 120
= 75 + 120
= 195
Since p(5) = 195 is positive, the inequality is not satisfied, and the club is not making a loss during this period either.
Finally, let's consider the interval (10, ∞). Choosing a value like t = 15:
p(15) = 15(15 - 10)(15 - 2) + 120
= 15(5)(13) + 120
= 975 + 120
= 1095
Since p(15) = 1095 is positive, the inequality is not satisfied, and the club is not making a loss during this period either.
Based on our calculations, the club did not make any losses during the years after the takeover, as the profit function remained positive for all values of t greater than 2.
Note: If the inequality had been satisfied for any interval, it would imply that the club was making a loss during that period. However, in this case, the inequality is not satisfied in any interval, indicating that the club did not experience any losses.