You make and sell birdhouses. Your
fixed costs for your tools and workspace are $3000. The cost of wood
and other materials needed to make a birdhouse is $10. You sell each
birdhouse for $50. Let x represent the number of birdhouses you make
and sell.
a. Write a function for your total costs, c(x).
b. Write a function for your income, i(x).
c. Analyze Your profit is the difference of your income and total costs.
Write a function for your profit, p(x).
d. What is your profit when you make and sell 100 birdhouses?
e. Interpret and Apply You are said to “break even” when your profit is
$0. How many birdhouses do you need to make and sell in order to
break even?
total costs c(x)= 3,000 + 10 x
i(x) = 50 x
p(x) = i(x) - c(x)
= (50-10)x - 3,000
= 40 x - 3,000
40(100) - 3,000 = 1,000
0 = 40 x - 3,000
x = 300/4 = 75
a. The total cost, c(x), is equal to the sum of the fixed costs and the variable costs.
Fixed costs: $3000
Variable costs: $10 per birdhouse
Therefore, the function for total costs is:
c(x) = 3000 + 10x
b. The income, i(x), is equal to the selling price of each birdhouse multiplied by the number of birdhouses sold.
Selling price: $50 per birdhouse
Therefore, the function for income is:
i(x) = 50x
c. The profit, p(x), is equal to the income minus the total costs.
Therefore, the function for profit is:
p(x) = i(x) - c(x)
= 50x - (3000 + 10x)
= 40x - 3000
d. To find the profit when making and selling 100 birdhouses, substitute x = 100 into the profit function:
p(100) = 40(100) - 3000
= 4000 - 3000
= $1000
e. To find the number of birdhouses needed to break even, we set the profit function equal to zero and solve for x:
p(x) = 0
40x - 3000 = 0
40x = 3000
x = 3000/40
x = 75
Therefore, you need to make and sell 75 birdhouses to break even.