To determine the number of units that result in a break-even situation for the product, which means the profit is zero, we need to solve the profit function [tex]\( P(x) \)[/tex] for when it equals zero. The profit function is given as:
[tex]\[ P(x) = -13x^2 + 1560x - 45,500 \][/tex]
The break-even points occur where [tex]\( P(x) = 0 \)[/tex]. Therefore, we need to solve the equation:
[tex]\[ -13x^2 + 1560x - 45,500 = 0 \][/tex]
This is a quadratic equation in the standard form [tex]\( ax^2 + bx + c = 0 \)[/tex], where:
- [tex]\( a = -13 \)[/tex]
- [tex]\( b = 1560 \)[/tex]
- [tex]\( c = -45,500 \)[/tex]
To solve this quadratic equation, we use the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Let's identify the discriminant first, which is the part under the square root:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Plugging in the values:
[tex]\[ \Delta = 1560^2 - 4(-13)(-45,500) \][/tex]
This calculation gives us the value of the discriminant [tex]\(\Delta\)[/tex]. Once we have [tex]\(\Delta\)[/tex], we can find the two solutions for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{-1560 \pm \sqrt{1560^2 - 4(-13)(-45,500)}}{2(-13)} \][/tex]
After calculating these values, we find two solutions for [tex]\( x \)[/tex], which indicate the break-even points. The break-even points are:
[tex]\[ x = 50 \][/tex]
[tex]\[ x = 70 \][/tex]
So, the number of units that give a break-even for this product are [tex]\( 50 \)[/tex] and [tex]\( 70 \)[/tex].
Therefore, the number of units that give the break-even for this product are [tex]\( 50, 70 \)[/tex].
