SOLUTION: Can someone help me out, please? Thank you! The owners of a parking garage have determined that their weekly revenue and cost in dollars are given by the following: R(x) =

Algebra ->  Functions -> SOLUTION: Can someone help me out, please? Thank you! The owners of a parking garage have determined that their weekly revenue and cost in dollars are given by the following: R(x) =       Log On


   

Question 153209: Can someone help me out, please? Thank you!

The owners of a parking garage have determined that their weekly revenue and cost in dollars are given by the following:
R(x) = 56x - 2x^2
C(x) = 24x + 101
find the break even point.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Remember,

Profit = Revenue - Cost

which is symbolically:


P%28x%29=R%28x%29-C%28x%29


P%28x%29=%2856x+-+2x%5E2%29-%2824x+%2B+101%29 Plug in R%28x%29+=+56x+-+2x%5E2 and C%28x%29+=+24x+%2B+101+


P%28x%29=56x+-+2x%5E2-24x+-+101 Distribute the negative.


P%28x%29=-2x%5E2%2B32x-101 Combine like terms


Now the break even point occurs when you neither gain money nor lose money. In other words, break even point happens when the profit is zero. So this means that the break even point occurs when P%28x%29=0


0=-2x%5E2%2B32x-101 Plug in P%28x%29=0


Notice we have a quadratic equation in the form of ax%5E2%2Bbx%2Bc where a=-2, b=32, and c=-101


Let's use the quadratic formula to solve for x


x+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29 Start with the quadratic formula


x+=+%28-%2832%29+%2B-+sqrt%28+%2832%29%5E2-4%28-2%29%28-101%29+%29%29%2F%282%28-2%29%29 Plug in a=-2, b=32, and c=-101


x+=+%28-32+%2B-+sqrt%28+1024-4%28-2%29%28-101%29+%29%29%2F%282%28-2%29%29 Square 32 to get 1024.


x+=+%28-32+%2B-+sqrt%28+1024-808+%29%29%2F%282%28-2%29%29 Multiply 4%28-2%29%28-101%29 to get 808


x+=+%28-32+%2B-+sqrt%28+216+%29%29%2F%282%28-2%29%29 Subtract 808 from 1024 to get 216


x+=+%28-32+%2B-+sqrt%28+216+%29%29%2F%28-4%29 Multiply 2 and -2 to get -4.


x+=+%28-32+%2B-+6%2Asqrt%286%29%29%2F%28-4%29 Simplify the square root (note: If you need help with simplifying square roots, check out this solver)


x+=+%28-32%2B6%2Asqrt%286%29%29%2F%28-4%29 or x+=+%28-32-6%2Asqrt%286%29%29%2F%28-4%29 Break up the expression.


So the answers are x+=+%28-32%2B6%2Asqrt%286%29%29%2F%28-4%29 or x+=+%28-32-6%2Asqrt%286%29%29%2F%28-4%29


which approximate to x=4.33 or x=11.67


So the break even points occur when the price is either $4.33 or $11.67