Answer:
a. The linear cost function is C(x) = 5.50x + 60.
b. Joanne must produce and sell 18 T-shirts in order break even.
c. Joanne must produce and sell 246 T-shirts to make a profit of $800.
I also showed how to find b. and c. just in case you wanted to see how to do these parts as well.
Explanation:
a. The general equation for cost (when it's linear) is
C(x) = mx + b, where
- C(x) is the cost to produce x units,
- m is the marginal cost (increase in cost per one additional unit produced),
- and b is the fixed cost (the cost a person must pay even when no units are produced)
We will need to find b (the fixed costs), by plugging in 390 for C(x), 5.50 for m, and 60 for x in the cost equation:
390 = 5.50(60) + b
390 = 330 + b
60 = b
Thus, the linear cost function is C(x) = 5.50x + 60.
b. There are two ways to find the break-even quantity:
- Find the profit function P(x), aka the difference of revenue and cost, set P(x) equal to 0, and solve for x.
- Set revenue equal to cost. This means set R(x) equal to C(x) and solve for x.
We know from a. that C(x) =5.50x + 60.
The linear revenue function is given by the equation R(x) = px, where
- R(x) is the revenue,
- p is the price of an item,
- and q is the quantity.
Because Joanne sells each silk-screened T-shirt for $9 per shirt, the revenue function is R(x) = 9x
Now we can find the difference between R(x) and C(x) to find P(x):
P(x) = R(x) - C(x)
P(x) = 9x - (5.50x + 60)
P(x) = 9x - 5.50x - 60
P(x) = 3.50x - 60
Thus, the linear profit function is P(x) = 3.50x - 60.
Now we can set P(x) equal to 0 and solve for x to find how many T-shirts Joanne must produce and sell in order to break even:
P(x) = 0
3.50x - 60 = 0
3.50x = 60
x = 17.14285714
x = 17
x = 18
Explaninig why x = 18:
The reason we had to round down to 17 initially is because you can't sell or produce part of a T-shirt. However, when we plug in 17 for x in P(x), you get -0.5 as
- P(17) = 3.50(17) - 60 = -0.5.
This means that Joanne can't break even by selling 17 units.
Thus, we must round to the next whole number which is 18. When you plug in 18 for x in P(x), you get 3 as
- P(18) = 3.50(18) - 60 = 3.
Although 3 is greater than 0, it's the number we must use.
c. In order to find how many T-shirts Joanne must produce and sell to make a profit of $800, we set P(x) equal to 800 and solve for x:
P(x) = 800
3.50x - 60 = 800
3.50x = 860
x = 245.7142857
x = 246
Explaining why x must be 246:
We know that Joanne can't sell or produce a fraction of a T-shirt. Thus, we mus round to the nearest whole number which is 246. When we plug in 246 for x, we get 801 as P(246) = 3.50(246) - 60 = 801. Although 801 is larger than 800, it's close enough to 800 that x = 246 is an acceptable answer: