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Problem 18 Medium Difficulty

The manager of a furniture factory finds that it costs\ $2200 to manufacture 100 chairs in one day and \$4800 to produce 300 chairs in one day.
(a) Express the cost as a function of the number of chairs produced, assuming that it is linear. Then sketch the graph.
(b) What is the slope of the graph and what does it represent?
(c) What is the y-intercept of the graph and what does it represent?

Answer

a. $C(N)=13 x+900$
b. The marginal cost of production is $\$ 13 /$ chair
c. 900

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Video Transcript

all right. This problem gives us a situation where there's a furniture factory and they're manufacturing chairs, and we want to express the cost as a function of the number of chairs produced, so cost will be our Y. Coordinate and number of chairs are X. I'm going to use end to stand for the number of chairs and see to stand for the cost. And they give us some data in the problem that we can translate into ordered pairs. For 100 chairs manufactured in a day, it costs $2200 and for 300 shares manufactured in a day, it costs $4800. So let's use those two points and find the equation of the line. And this is part A. So we can start by finding the slope changing. Why 4800 minus 2200 over Change in x 300 minus 100 gives us 2600 divided by 200 that's 13 and then we can use our point slope form. Why minus y one equals M times X minus X one. We can use one of our two points. It doesn't matter which, so I'll just use the 1st 1 Why minus 2200 equals 13 times X minus 100 and we can simplify that. First will distribute the 13. So why minus 2200 equals 13 x minus 1300 and then we'll add 2200 to both sides. So why equals 13 X plus 900? Now if I want to put it into the variables that I chose instead of X and Y will say that, see, the cost equals 13 times and the number of chairs plus 900. So there's our linear model, and now we can make a sketch of it. So let's assume that they're not going to make a negative number of chairs. So we'll just focus on quadrant one number of chairs on X cost on Why go buy one hundreds on the X axis and I'll go buy one thousands on the Y axis. Now, remember, we just found our equation that has a Y intercept of 900 so we can pull at that point and we have the 0.122 100 and we have the 0.348 100 and those should fall in a line. And so there's a sketch of the graph. All right, The next thing we want to look at is what is the slope and what does it represent? So if we look back at the function we found the slope was 13. And if you think about the units the units on, why was ah, the cost? So that would be dollars and the units on X, that was the number of chairs, so dollars per chair. So it's $13 more for every chair increased cost $13 per chair. Okay, how about the Y intercept? We can see from our equation that the Y intercept is 900. That would correspond to the 0.0 comma. 900 and zero would be the number of chairs made. So if they make zero chairs, they still have a cost of $900 per day at their manufacturing plant.

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