Question

The revenue for a company producing friendship bracelets is given f(x) = -15x^2 - 30x + 200, where x is the price in dollars for each friendship bracelet. The cost for the production is given by y = 5x - 12. What price should the company sell the friendship bracelets to break even?

Answer 1

Revenue is given by f(x) = -15x^2 - 30x + 200

Cost is given by y = 5x - 12

to break even, then Revenue = Cost

Then:

`-15x^2-30x+200=5x-12`

Now, let's solve for x, which is the price

Step #1.

add 12 to both sides

`-15x^2-30x+200+12=5x-12+12`

#2

Simplify

`-15x^2-30x+212=5x`

#3

subtract 5x from both sides

`-15x^2-30x+212-5x=5x-5x`

#4

Simplify

`-15x^2-35x+212=0`

#5

Use the general formula to solve quadratic equation

`x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a};a^2x+bx+c=0`
`x_(1,\: 2)=(-\left(-35\right)\pm√(\left(-35\right)^2-4\left(-15\right)\cdot\:212))/(2\left(-15\right))`

#6

`x_1=(-\left(-35\right)+√(13945))/(2\left(-15\right)),\: x_2=(-\left(-35\right)-√(13945))/(2\left(-15\right))`

#7

`x=-(35+√(13945))/(30),\: x=(√(13945)-35)/(30)`

#8

in decimal numbers

`x=-5.10,\: x=2.77`

a negative value does not make sense in this case, therefore the answer is

d) $2.77