SOLUTION: I am usually pretty good at word problems but this one is stumping me. Sharon's revenue $R on the sale of fruitcakes is determined by the formula R=50x-x^2. Her cost $C for p

Algebra ->  Customizable Word Problem Solvers  -> Finance -> SOLUTION: I am usually pretty good at word problems but this one is stumping me. Sharon's revenue $R on the sale of fruitcakes is determined by the formula R=50x-x^2. Her cost $C for p      Log On

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Question 1932: I am usually pretty good at word problems but this one is stumping me.
Sharon's revenue $R on the sale of fruitcakes is determined by the formula R=50x-x^2. Her cost $C for producing x fruitcakes is given by the formula C=2x+40. For what values of x is Sharon's profit positive?
Can you please help??
Found 2 solutions by longjonsilver, khwang:
Answer by longjonsilver(2297) About Me  (Show Source):
You can put this solution on YOUR website!
this is a very good question as it uses graphs to highlight a realistic problem.
Basically, what it is asking you is where is her revenue larger than her costs. You could draw these graphs together and then you can see where they intersect with each other. However, this is tedious and not very precise.
Much better to combine the 2 equations and find the roots.
Basically you are asking..."where are these 2 equations the same", which is what you ask when a question says "solve"
so, 50x+-+x%5E2+=+2x+%2B+40
Re-arranging gives x%5E2+-+48x+%2B+40+=+0
This is not factorisable, so use the formula. Doing this gives
x+=+%2848+%2B-+sqrt%282144%29%29%2F%282%29
so either work this out or simplify as...
x+=+%2848+%2B-+sqrt%28%28134%29%2816%29%29%29%2F%282%29
x+=+%2848+%2B-+sqrt%28134%29sqrt%2816%29%29%2F%282%29
x+=+%2848+%2B-+4sqrt%28134%29%29%2F%282%29
x+=+24+%2B+2sqrt%28134%29 or x+=+24+-+2sqrt%28134%29
Any x-value between these 2 roots will be where revenue surpasses costs...Sketch the 2 graphs to prove this to yourself.
Jon

Answer by khwang(438) About Me  (Show Source):
You can put this solution on YOUR website!
Profit = R -C = 50x -x^2 - (2x+40)
= 48 x -x^2 - 80
= -(x^2 - 48x + (24)^2) + 576 - 80
= - (x -24)^2 + 496
>= 0
(x -24)^2 < 496
|x -24| < sqrt(496) = 22.27
-22.27 + 24 < x < 22.27 + 24
1.73 < x < 46.27
Since x must be integers, so 2 <= x <= 46.
Answer: When producing x from 2 to 46 (2 <=x <= 46)cakes,
the profit is positive.

Kenny