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 groph. (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
groph.
(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

00:03 We are given the cost of production of a product, and we are given the amount of product that can be produced in a single day and how they are related.

00:22 And we are asked in part a to determine a formula relating these two quantities.

00:30 So we also assume that this function is going to be linear.

00:38 So let's call the cost of manufacturing x chairs per day cx and x is the number of chairs manufactured per day.

01:04 Then we see that c of 100 is equal to 2 ,200, and c of 300 of 300 is equal to 300 and c of 300 is equal to 4 ,800.

01:32 And we're assuming that c is a linear function of x.

01:37 So c of x is equal to mx plus b for some n and b in the real numbers.

01:45 Therefore, plugging in some of these values, we get that c of 100 is equal to 100m plus b, which is equal to 2 ,200, and we get that c of 300 is equal to 300m plus b, which is equal to 4 ,800.

02:14 Simply subtracting the top equation from the bottom, we get the 200m is equal to 4800 minus 2200 is 2 ,600.

02:27 So that m is equal to 26 over 2 or 13.

02:37 So it follows that c.

02:41 Of 100 is equal to 100 times 13 plus b, which is equal to 2 ,200, so that b is equal to 2 ,200 minus 1 ,300 or 900.

03:05 Therefore, c of x is equal to 13x plus 900.

03:17 This is the answer for part a.

03:23 In part b, we are asked to interpret the graph.

03:34 So first, i'm going to sketch the graph.

03:40 Now, of course, you can't produce negative chairs per day.

03:45 So x is going to be greater than equal to zero.

03:49 And the cost of producing chairs is always going to be greater than equal to zero.

03:55 I don't think this problem is accounting for any sort of subsidies or anything.

04:00 So the cost is always going to be positive, or at least non -negative...

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