Ex: Find Total Revenue, Total Cost, and Total Profit Functions

The demand for our product is given by the following demand function, d of q = -1.5q + 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 fix cost of production for this item or $5,000 and the variable costs are $9 per item produced. If 27 items are produced and sold find the following: a, the total revenue from selling 27 items, b, the total cost to produce 27 items, and, c, the total profits to produce and sell 27 items.

So for part a we want to find the total revenue from selling 27 items. Where the revenue is equal to the quantity of items sold, which is 27 x 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, again, the quantity x the price. And, again, the price is d of q, so we can write this as q x d of q, or just q x the quantity -1.5q + 290.

So if we distribute here, notice the revenue function r of q is a quadratic function. We have -1.5q squared + 290 x q, which means r of 27 or the total revenue, when q = 27 is -1.5 x 27 squared + 290 x 27.

So -1.5 x 27 squared + 290 x 27 gives us the total revenue of $6,736.50. Other option here would've been to find p by determining d of 27, and then multiplying 27 x the price as we see here.

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.

Well, the total costs consist of the fixed cost and the variable cost. So the total cost c of q would be equal to the fixed cost of $5,000 + the variable cost of $9 per item.

So we'd have + 9 x q. And, therefore, the total cost to produce 27 items would be c of 27, which would be 5,000 + 9 x 27. 9 x 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 the total cost of c of q.

So if q = 27 the total profit would be p of 27, which is equal to r of 27 c of 27. Well, we just found these two function values, r of 27 was equal to 6,736.5, and c of 27 was equal to 5,243.

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