Worked example for working out break-even point from Cost and Revenue functions.
The cost function for flavored coffee at an upscale coffeehouse is given in dollars by C(x) = 3x + 160, where x is in pound. then coffee sells for $7 per pound.
A. find the break-even quantity.
B. what will the revenue be at the point?
C. what is the profit from 100 pounds?
D. how many pound of coffee will produce a profit of $500?
QUESTION:
The cost function for flavored coffee at an upscale coffeehouse is given in dollars by C(x) = 3x + 160, where x is in pound. then coffee sells for $7 per pound.
A. find the break-even quantity.
B. what will the revenue be at the point?
C. what is the profit from 100 pounds?
D. how many pound of coffee will produce a profit of $500?
Worked example:
Cost (in dollars) for an enterprise producing salted pistachios is given by C(x) = 0.25x + 180
where x is in ounces.
This equation tells us there is a fixed running cost of $180 in addition to the quarter it costs to produce each pistachio.
A 6 ounce bag of salted pistachios sells for $9 therefore sale price is $1.50/ounce
Revenue (in dollars) from producing salted pistachios is given by R(x) = 1.5x
When the revenue received by the enterprise is greater than the cost of production
it makes a profit {Profit = R(x) - C(x)}
If the revenue is less than the cost a loss is made {Loss = C(x) - R(x)}
When the revenue is the same as the cost the enterprise breaks even {R(x) = C(x)}
* To find the break-even quantity we are working out how many ounces of pistachios the enterprise must sell before it starts to make a profit.
We know cost is given by C(x) = 0.25x + 180 and revenue is given by R(x) = 1.5x
Break-even point occurs when C(x) = R(x) Equating both equations gives: 0.25x + 180 = 1.5x
⇒ 180 = 1.25x ⇒ 720 = 5x ⇒ x = 144
This tells us the enterprise must sell 144 ounces of pistachios to break-even.
Each bag sold weighs 6 ounces therefore 24 bags of pistachios must be sold before making a profit.
* At break-even point we know x = 144. Revenue is given by R(x) = 1.5x
It follows R(144) = 1.5(144) = $216
* We can now work out the profit from selling 150 bags of pistachios. Number of ounces of pistachios sold = 6 X 150 = 900
Profit = R(x) - C(x) When x = 900, Profit = R(900) - C(900) = 1.5(900) - [0.25(900) + 180)]
⇒ Profit = 1350 - (225 + 180) = 1350 - 405
⇒ Profit = $945
* We can also work out how many bags of pistachios need to be sold to make a profit of $1500.
We know Profit = R(x) - C(x) where Profit = 1500, R(x) = 1.5x, C(x) = 0.25x + 180
Therefore 1500 = 1.5x - (0.25x + 180) ⇒ 1500 = 1.25x - 180 ⇒ 1.25x = 1680 ⇒ 5x = 6720
⇒ x = 1344 1344/6 = 224
The enterprise must sell 224 bags of pistachios to make a profit of $1500.
