Question 1191483: The manufacturer of a water bottle spends $5 to build each bottle and sells them for $10. The manufacturer also has fixed costs each month of $9,500.
(a)
Find the cost function C when x bottles are manufactured.
C(x) =
(b)
Find the revenue function R when x bottles are sold.
R(x) =
(d)
Find the break-even point. Interpret what the break-even point means.
When___water bottles are sold, the cost and revenue equal $___.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! **Cost Function (C(x))**
* Variable cost per bottle: $5
* Fixed costs: $9,500
* Number of bottles manufactured: x
C(x) = (Variable cost per bottle * Number of bottles) + Fixed costs
**C(x) = 5x + 9500**
**Revenue Function (R(x))**
* Selling price per bottle: $10
* Number of bottles sold: x
R(x) = Selling price per bottle * Number of bottles sold
**R(x) = 10x**
**Break-Even Point**
The break-even point is where the cost equals the revenue (C(x) = R(x)).
1. Set the cost function equal to the revenue function:
5x + 9500 = 10x
2. Solve for x:
9500 = 5x
x = 1900
3. Calculate the cost/revenue at this point:
C(1900) = 5 * 1900 + 9500 = $19,000
R(1900) = 10 * 1900 = $19,000
**Interpretation**
**When 1900 water bottles are sold, the cost and revenue both equal $19,000.** This means the manufacturer has covered all their costs (both fixed and variable) and has not yet made a profit. Any bottles sold beyond 1900 will generate profit.
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