Answer:
1. 250 = 13q - 60
2. 23.84615385 = q
The solution means that if it were possible for Susan to sell 23.84615385 purses exactly, she'd make a profit of $250 exactly. However, because Susan can't sell a fraction of a purse, she'd need to sell at least 24 purses, which would make her profit slightly more than $250, thereby allowing her to both meet and exceed her profit goal.
3. We determined in the second item that the least amount of purses Susan must sell to meet her profit goal of $250 is 24 purses. Selling only 20 purses therefore would cause Susan to not meet her profit goal.
Explanation:
The problem requires us to the use the revenue, cost, and profit functions.
General equation for the revenue function and the revenue function for Susan:
Revenue is the product of price and quantity:
Revenue = price * quantity
Thus, the revenue function is given by:
R(q) = pq, where
- R is the revenue per q units sold,
- and p is the price of an item.
Since Susan sells each purse for $25, we can model her revenue with the following equation:
R(q) = 25q
General equation for the cost function and the cost function for Susan:
Cost is the sum of the variable cost (marginal cost * quantity) and the fixed cost:
Cost = variable cost + fixed cost
Thus, the cost function is given by:
C(q) = mq + b, where
- C is the cost per q units made,
- mq is the marginal cost (increase in cost per each additional unit made),
- and b is the fixed cost (the cost paid even if no units are made.
Since it costs Susan $12 to make each purse and she rents the booth for $60, we can model her cost using the following equation:
C(q) = 12q + 60
General equation for the profit function and the cost function for Susan:
Profit is the difference of revenue and cost:
Profit = Revenue - Cost
Thus, the profit function is given by:
P(q) = R(q) - C(q), where
- q is the quantity of items.
Finding the profit function for Susan:
Now we can find the profit function for Susan by subtracting her cost function from her revenue function:
P(q) = 25q - (12q + 60)
P(q) = 13q - 60
Now we can begin answering the questions:
1.
Since P(q) represents the profit and we want to determine the amount of purses Susan must sell to make a profit of $250, we can do so using the following equation:
250 = 13q - 60
2.
Now we want to solve the equation by isolating q:
(250 = 13q - 60) + 60
(310 = 13q) / 13
310/13 = q
23.84615385 = q
The solution means that if Susan were to sell 23.84615385 purses exactly, she'd make a profit of $250 exactly. However, because Susan can't sell a fraction of a purse, she'd need to make and sell at least 24 purses, which would make her profit slightly more than $250, thereby allowing her to both meet and exceed her profit goal.
3.
We determined in the second item that the least amount of purses Susan must sell to meet her profit goal of $250 is 24 purses. Selling only 20 purses would cause Susan to not meet her profit goal.