The manager of a furniture factory finds that it costs 2200...

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?

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?

Key Concepts

Linear Function

A linear function represents a relationship between two variables that produces a straight-line graph. It is typically written in the form y = mx + b, where 'm' represents the constant rate of change (slope) and 'b' represents the y-intercept, the value of y when x is zero.

Cost Function

A cost function is a mathematical model that shows the relationship between the cost incurred and the number of items produced. Often, it includes a fixed cost component and a variable cost per unit, which together can be modeled using a linear function when the relationship is linear.

Slope

The slope in a linear function indicates the rate at which the dependent variable changes with respect to the independent variable. In the context of a cost function, the slope represents the additional cost incurred for producing one more unit, essentially the variable cost per item.

Y-intercept

The y-intercept of a linear function is the value of the function when the independent variable equals zero. In cost functions, this is typically interpreted as the fixed cost, which is the cost incurred even if no items are produced.

Transcript

00:01 Ok for this problem we're given that a furniture factory finds it's $2 ,200 to manufacture 100 chairs and $4 ,800 to produce 300 chairs.

00:11 The first thing i did was jot down those two pieces of information and put them as an ordered pair where x is representing the number of chairs and y is representing the cost, the total cost.

00:28 So it says express the cost as a function of the number of chairs produced.

00:32 So for a it's saying the cost, the number of chairs, so let's do cost, let's call it c actually now that we're doing that.

00:41 So i'm going to put, this is still like an xy coordinate, but let's go ahead and do cost of x is equal to the number of chairs produced.

00:52 So we need to figure out what the slope is of this function.

00:57 So let's go ahead and do that.

00:59 So slope is rise over run, so the change in y over the change in x.

01:06 200 on the bottom and the numerator we have 2 ,600.

01:12 Simplifying that down it's 13, so our slope is 13.

01:17 We can now use that to calculate our y intercept.

01:22 So we know it's 13x plus b, b being our y intercept.

01:26 So we're going to go ahead and use one of the ordered pairs and use the smaller of the two, so 2 ,200 is equal to 13 times 100 plus this intercept.

01:38 It's going to be likely a negative number and it's not actually a useful intercept for us because we're never going to produce zero chairs.

01:47 That's just going to say how much money they lose each day.

01:49 This is 1 ,300, let's see, 1 ,300 plus b, it looks like they're not losing money.

01:58 So b here is going to be 900.

02:01 So it sounds like if they don't produce anything they're still earning $200, which is a little weird.

02:06 So our cost function is c of x is equal to 13x plus 900.

02:13 So for part a it asks us to express the cost of function of the number of chairs produced, this means linear, then sketch the graph.

02:21 So in a graph here we just need the first quadrant and so we can go ahead and create the metrics here...

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