Question

Skylar repairs microwaves. Her revenue, in dollars, can be modeled by the equation y = 80 + 30x, where x is the number of hours spent repairing microwaves. Her overhead cost, in dollars, can be modeled by the equation y=10x2−20 , where x is the number of hours spent repairing microwaves. After how many hours does she break even?

Answer 1

`x=5` 5 hours.

Explanation:

We have been given an equation
`y=80+30x`, which represents the revenue in dollars for Skylar.

The equation
`y=10x^2-20` represents Skylar's overhead cost.

Since we know that break even is the point, where total cost equals to total revenues, so to find the break-even we will equate our given both equations as:

`10x^2-20=80+30x`

Now let is solve for x.

`10x^2-20-80-30x=80-80+30x-30x`

`10x^2-30x-100=0`

Let us divide our equation by 10.

`x^2-3x-10=0`

Let us factor out our quadratic equation by splitting the middle term.

`x^2-5x+2x-10=0`

`x(x-5)+2(x-5)=0`

`(x+2)(x-5)=0`

`(x+2)=0\text{ or }(x-5)=0`

`x+2-2=0-2\text{ or }x-5+5=0+5`

`x=-2\text{ or }x=5`

Since the number of hours can not be negative, therefore, the break-even point is 5 hours.

Answer 2

after how many hours does she break even?
y = 80 + 30x=10x2−20
so 10x²-30x-100=0 or x²-3x-10=0
Delta= 3²-4.(1).(-10)=9+40=49
the root of Delta is sqrt49=7
so x= 3-7/2= -2 or x=3+7/2=5
so the number of hours is after x=5 hours