The profit P(x) (in millions of dollars) for a T-short manufacturer can be modeled by P(x) = -2x^3+9x^2-x, where x is the number of T-shirts produced (in millions). Currently, the company produces 4 million T-shirts and makes a profit of $12 million. What lesser number of T-shirts could the company produce and still make the same profit?
You just want to find for what value of x we get
-2x^3 + 9x^2 - x = 12 , or
2x^3 - 9x^2 + x + 12 = 0
You already told me that x = 4 would be a solution, so x - 4
must be a factor.
Use either synthetic or long algebraic division to show that
(x - 4)(2x^2 - x - 3) = 2x^3 - 9x^2 + x + 12
so we get two more solutions from 2x^2 - x - 3 = 0
which factors to (x+1)(2x-3) = 0
x = -1, or x = 3/2, of course we have to reject the x = -1, (not good business
practise to have negative sales)
so when they sell 3/2 or 1.5 million shirts they make the same profit
check:
-2(1.5)^3 + 9(1.5)^2 - 1.5 = 12
To find the lesser number of T-shirts the company could produce and still make the same profit, we need to solve the equation P(x) = 12, where P(x) is the profit given by the equation P(x) = -2x^3 + 9x^2 - x.
Substituting P(x) = 12 into the equation, we get:
12 = -2x^3 + 9x^2 - x
Rearranging the equation to a standard form of a cubic equation:
2x^3 - 9x^2 + x - 12 = 0
Now, we need to solve this equation to find the value(s) of x.
To solve it, you can try using numerical methods or a graphing calculator. The solutions to this equation will give us the number of T-shirts that will yield the same profit as the current scenario.
To find the lesser number of T-shirts that would yield the same profit, we need to set up an equation and solve for x.
Given that the profit function is P(x) = -2x^3 + 9x^2 - x, and the current profit is $12 million when 4 million T-shirts are produced, we can substitute these values into the equation.
P(4) = 12
So, we plug in x = 4 into the profit function and solve for P(x):
P(4) = -2(4)^3 + 9(4)^2 - 4
P(4) = -2(64) + 9(16) - 4
P(4) = -128 + 144 - 4
P(4) = 12
Since the current profit is $12 million, we have verified that our equation is correct.
Now, we want to find a lesser number of T-shirts that would yield the same profit, so we need to solve the equation P(x) = 12 for x.
-2x^3 + 9x^2 - x = 12
Rearranging the equation:
-2x^3 + 9x^2 - x - 12 = 0
To solve this polynomial equation, we can use either factoring, the rational root theorem, or numerical methods. In this case, let's try factoring the expression.
By trial and error, we can find that x = 3 is a solution. Hence, we can factor out (x - 3):
(x - 3)(-2x^2 - 3x - 4) = 0
To find the remaining factor, which is a quadratic expression, we can either use factoring, completing the square, or the quadratic formula.
Using the quadratic formula, we have:
-2x^2 - 3x - 4 = 0
x = (-(-3) ± √((-3)^2 - 4(-2)(-4))) / (2(-2))
x = (3 ± √(9 - 32)) / (-4)
x = (3 ± √(-23)) / (-4)
Since the value inside the square root is negative, this equation does not have any real solutions. Therefore, the only solution is x = 3.
Therefore, the company could produce a lesser number of T-shirts, namely 3 million T-shirts, and still make the same profit of $12 million.