To solve the problems given in the quadratic profit function \( f(p) = -50p^2 + 1600p - 11,000 \), we can identify a quadratic equation in standard form, \( f(p) = ap^2 + bp + c \), where \( a = -50 \), \( b = 1600 \), and \( c = -11,000 \).
(a) Find the price that generates the maximum profit.
The maximum profit for a quadratic function can be found at the vertex, given by the formula \( p = -\frac{b}{2a} \).
Substituting the values of \( a \) and \( b \):
\[ p = -\frac{1600}{2 \cdot -50} = -\frac{1600}{-100} = 16 \]
Thus, the price that generates the maximum profit is \( p = 16 \) dollars per shirt.
(b) Find the maximum profit.
To find the maximum profit, we substitute \( p = 16 \) back into the profit function \( f(p) \):
\[ f(16) = -50(16^2) + 1600(16) - 11,000 \]
Calculating \( 16^2 \):
\[ 16^2 = 256 \]
Now substituting this back:
\[ f(16) = -50(256) + 1600(16) - 11,000 \] \[ = -12800 + 25600 - 11000 \] \[ = 12800 - 11000 \] \[ = 1800 \]
Thus, the maximum profit is \( 1800 \) dollars.
(c) Find the price(s) that would enable the company to break even.
To find the break-even prices, we need to set the profit function equal to zero:
\[ -50p^2 + 1600p - 11,000 = 0 \]
We can use the quadratic formula \( p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = -50 \), \( b = 1600 \), and \( c = -11,000 \).
First, calculate the discriminant:
\[ b^2 - 4ac = 1600^2 - 4(-50)(-11000) \] \[ = 2560000 - 2200000 \] \[ = 360000 \]
Now, substituting this into the quadratic formula:
\[ p = \frac{-1600 \pm \sqrt{360000}}{2 \cdot -50} \] \[ = \frac{-1600 \pm 600}{-100} \]
Now, we find the two possible values for \( p \):
- \( p_1 = \frac{-1600 + 600}{-100} = \frac{-1000}{-100} = 10 \)
- \( p_2 = \frac{-1600 - 600}{-100} = \frac{-2200}{-100} = 22 \)
Thus, the prices that would enable the company to break even are \( p = 10 \) and \( p = 22 \).
So, the answers to the parts are:
- (a) The price that generates the maximum profit is $16.
- (b) The maximum profit is $1800.
- (c) The break-even prices are $10 and $22.
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