The unit cost, in dollars, to produce tubs of ice cream is...

The unit cost, in dollars, to produce tubs of ice cream is $12 and the fixed cost is $5562. The revenue function, in dollars, is R(z) = -2x^2 + 272x Find the profit function. P(z) = At what quantity is the smallest break-even point?

Transcript

00:01 So for this question, we are given a revenue function, which i have labeled as r of x, and we're also told some information about the cost.

00:09 We know that it's $12 per unit, so that's why you have 12x in the cost function.

00:16 You're also told that there's a fixed cost of $5 ,562, so that has to get added on as well.

00:23 Now what they want for this question is first we need to figure out a profit function.

00:28 Profit is equal to the money you make minus however much you have to spend.

00:35 So it's your revenue minus your cost.

00:39 So you can substitute in each of these functions.

00:42 So we have revenue, and then it's minus cost.

00:53 The tricky part here is you have to be careful to put your cost function in parentheses.

01:02 So let's go ahead and simplify this out.

01:05 So we have our negative 2x squared.

01:07 We will have 272x minus 12, so that's plus 260x, and then minus 5 ,562.

01:22 So this right here is the profit function.

01:28 Now there's a second part to this question that wants to know how many units need to produce in order to break even.

01:37 Now when they say break even, that means that you make no profit.

01:43 So you set your profit function equal to zero and solve.

01:48 So let's go ahead and do that.

01:50 If i want to solve this, it's a quadratic equation, i'd probably want to divide both sides by negative 2 first...

Question

The unit cost, in dollars, to produce tubs of ice cream is $12 and the fixed cost is $5562. The revenue function, in dollars, is R(z) = -2x^2 + 272x Find the profit function. P(z) = At what quantity is the smallest break-even point?

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