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Question 530290: I still don't get it, please help me.
A firm has a fixed cost of $28000. For every unit of the product it makes, the cost increases by 0.4x +222$ where x is the number of units produced. The selling price of the product is given by the function p(x) = 1250 - 0.6x. What is the maximum number of units the firm can make and sell, and still make a profit?
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! I'm not into economics, but since no other tutor has tried to help you with this, I'm going to take a shot at it, and together we'll see if we can make some sense out of this problem.
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In an enterprise, a manufacturer usually (not always, though) has the goal of making money. That is to say the manufacturer wants the income it gets from producing a product to be more than the cost of making the product. The difference between the income and the cost is called profit. If profit is positive, the firm is making money. If profit is zero, the firm is breaking even. And if profit is negative, the firm is losing money. Seems pretty basic, but we're going to use this idea to solve (hopefully) this problem.
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We're told that the firm has a fixed cost of 28,000 dollars. Fixed costs are costs that in general do not change regardless of how many units the firm produces. Such fixed costs may include things like paying taxes on the property, doing basic maintenance needed to prevent the property from deteriorating, providing a basic amount of heating and air conditioning so a certain amount of people can work in the place, etc.
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But in addition to that, the firm has found that there is a cost of production that increases with each unit it produces to sell. Somebody at the firm has figured out how to model or calculate this total cost. If given the total units produced (call the unknown number of units "x"), the total variable cost can be calculated from the relationship 0.4x + 222 dollars. Notice that this relationship is in the algebraic form of a slope-intercept one. That tells us its graph will be a straight line, with a positive slope of +0.4 and a y-intercept of 222. That means that for each increase of 1 unit, this total variable cost will go up by 0.4 dollars (or 40 cents).
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The total cost of production will be the sum of the fixed costs plus the variable costs. Therefore, we can say that the total cost (call it "C") of production is:
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C = 28000 + 0.4x + 222
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Combine the two constants on the right side and this cost equation becomes:
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C = +0.4x + 28222
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This equation also is in the slope intercept form. On the right side we can see that for each unit that we increase x, the cost goes up 0.4 dollars, but the constant means that that the firm has a fixed cost of 28,222 dollars regardless of how many units are produced. Even if x equals zero (no units produced), the fixed cost of running the place is $28,222.
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Now that we have a feel for the cost side of the problem, let's look at the income side.
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We are told that the selling price of this product (in dollars) is (rearranging the terms a little):
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p(x) = -0.6x + 1250
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Note that this is the price per unit, not the total income. You can tell because if it were total income, the most the firm could make would be $1250 and that would be by producing zero units. After that, each additional unit would reduce the total income. Not likely. Nope. This equation tells us that the most that a customer could be charged for a unit is $1250 (in the unlikely case that zero units are made). After that, presumably due to such things as the increasing skill of the workers as their experience increases and the price breaks the firm gets for buying larger quantities of raw materials, the cost per unit goes down at a rate of 0.6 dollars (60 cents) per unit produced.
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So, how would you determine the total income. Relatively easy. Let "I" represent total income. Then "I" would be equal to the number of units produced times the price of each unit. So we can say:
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I = x*p(x) = x*(-0.6x + 1250)
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Multiplying out the right side results in the equation:
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I = -0.6x^2 + 1250x
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The right side of this equation is a quadratic relationship, not a linear one. Its graph will be a parabola. It tells us the amount of the total income the firm can expect to receive by selling "x" number of units.
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Now let's go back to our original discussion. The amount of profit (P) that the firm will get can be found by subtracting the cost (C) of "x" units from the amount of income (I) from selling "x" units. So:
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P = I - C
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Let's substitute the relationships that we developed for I and C to get:
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P = (-0.6x^2 + 1250x) - (0.4x + 28222)
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Remove the parentheses and you have:
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P = -0.6x^2 + 1250x - 0.4x - 28222
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Combine the two terms containing "x" and the equation becomes:
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P = -0.6x^2 + 1249.6x - 28222
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This is a quadratic equation. If it has real solutions, it should give us a domain for x (or possibly domains) over which P is positive, and our goal is to find where P is positive. So let's set P equal to zero (the break even point) and find out over what span the value of P is positive. If we set P equal to zero and use it as the right side of the equation we get:
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-0.6x^2 + 1249.6x - 28222 = 0
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Hopefully you know how to use the quadratic formula to solve this equation. To save time in answering this problem, I'll tell you that the two answers are x = 2059.83 and x = 22.8352.
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Where are we now? In the way of explanation, we know that the cost equation is linear and has straight line graph. And the income equation was quadratic and has a parabolic graph. We've just solved for the values of "x" where the two graphs are equal, that is to say, we found the two values of x where the difference between total income and total cost is zero. This will be where the two graphs cross. The graphs look like this:
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Note: The x-axis represents the units that are produced, and the y-axis shows the dollars. Because of the scale of this graph, the red line graph looks like a horizontal line. It isn't. It is the Cost graph, and it crosses the vertical y-axis at $28222 and (as you look to the right) it slopes upward at a rate of +0.4 dollars per unit increase in x. The green graph shows the amount of income. It is a quadratic/parabola graph of our income model represented as x*(-0.6x + 1250).
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Also note: on its way up the green graph crosses the red graph where x equals 22.8352 and again on its way down where x = 2059.83. Between these two values of x notice that the green graph is higher than the red graph, meaning that the green graph (total income) is higher than the red graph (total cost). Therefore between the values of x = 23 (rounded to the next integer up) and x = 2059 (rounded to the next integer down) the total income is more than the total cost and the company is making money.
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Therefore, we have answered the question. The maximum number of units the firm can make and still have some profit is 2059. At 2060 units the firm will lose a little (very little) money because the green graph (total income) is slightly below the red graph (total cost). This means that total income minus total cost is negative. And for each additional unit made the total loss (negative profit) just gets to be more and more.
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As an observation, note where the company makes the most money. It is around where the green curve reaches its peak. Therefore, the firm needs to consider making only the number of units (the value of x) around where the green curve reaches a maximum. After that, adding more units just reduces the total profitability.
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Hope this answer helps you to understand the problem a little bit more.
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