The profit for a product is given by $\mathrm { p } = 2204 + \mathrm { x } ^ { 2 } - 105 \mathrm { x }$ x, where x is the number of units produced and sold. Graphically find the x-intercepts of this function to find how many units will give break-even (that is return a
Profit of zero).
A) 29 units
B) 76 units
C) 29 or 76 units
D) They will never break even.

Question

Quizplus · Accepted Answer

To find the x-intercepts (break-even points), set the profit function to zero and solve for x: $0 = 2204 + x^2 - 105x$. This is a quadratic equation, which can be solved either by factoring, completing the square, or using the quadratic formula. The quadratic formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a = 1$, $b = -105$, and $c = 2204$. Plugging these values into the formula gives two solutions for x, which represent the break-even points. The calculation will show that the break-even points are at 29 units and 76 units, meaning the company will break even if they sell either 29 units or 76 units.

The Profit for a Product Is Given By p=2204+x2−105x\mathrm { p } = 2204 + \mathrm { x } ^ { 2 } - 105 \mathrm { x }p=2204+x2−105x

The Profit for a Product Is Given By $\mathrm { p } = 2204 + \mathrm { x } ^ { 2 } - 105 \mathrm { x }$