The demand for our product is given by the following demand function:
D of Q equals negative 1.5 Q plus 290, where Q is the quantity of items in demand,
and D of Q is the price per item in dollars that can be charged when Q items are sold.
Suppose the fixed costs of production for this item are $5,000, and the variable
costs are $9 per item produced. If 27 items are produced and sold, find the following:
Part A: The total revenue from selling 27 items. The revenue is equal to the quantity of items
sold, which is 27, times the price of the item, which we’re not given. But notice how we are given
the demand function, where D of Q would be the price per item when Q units are sold. So R of Q
would be equal to the quantity times the price, but the price is D of Q. So we can write this as Q
times D of Q, or just Q times the quantity negative 1.5 Q plus 290.
So if we distribute here, notice the revenue function R of Q is a quadratic function.
We have negative 1.5 Q squared plus 290 times Q, which means R of 27, or the total revenue when Q
equals 27, is negative 1.5 times 27 squared plus 290 times 27. So negative 1.5 times 27 squared
plus 290 times 27 gives us the total revenue of $6,736.50.
Our other option here would have been to find P by determining D of 27,
and then multiplying 27 times the price, as we see here. But later on,
we are going to need the revenue function in terms of Q, as we see here in quadratic form.
Now for part B, we were asked to find the total cost to produce 27 items, where the
total cost consists of the fixed costs and the variable cost. So the total cost, C of Q, would
be equal to the fixed cost of $5,000, plus the variable cost of $9 per item. So we'd have plus
9 times Q. Therefore, the total cost to produce 27 items would be C of 27, which would be 5,000
plus 9 times 27. 9 times 27 is 243, so the total cost to produce 27 items is $5,243.
And now for part C, we're asked to find the total profits to produce and sell 27 items. Well, the
total profits, P of Q, would be equal to the total revenue, R of Q, minus the total cost, C of Q.
So if Q equals 27, the total profit would be P of 27, which is equal to R of 27 minus C of 27.
Well, we just found these two function values. R of 27 was equal to $6,736.50,
and C of 27 was equal to $5,243.
This difference is $1,493.50, which means the total profit is $1,493.50.
I hope you found this helpful.