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? (b) 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?
(b) What is the $y$-intercept of the graph and what does it represent?

Key Concepts

Linear Function

A linear function is an algebraic expression of the form y = mx + b that represents a straight-line graph. It models a relationship where the rate of change is constant. In cost analysis, it is typically used to represent the total cost as the sum of a fixed component and a variable cost that depends on the quantity produced.

Slope

The slope of a linear function is the coefficient of the independent variable in the equation, representing the rate of change of the dependent variable with respect to the independent variable. In cost functions, the slope often signifies the variable cost per unit, indicating how much the total cost increases with an additional unit produced.

Y-Intercept

The y-intercept is the value of the dependent variable when the independent variable is zero, found as the constant term in the linear equation. In the context of cost functions, it represents the fixed costs that are incurred even when production is zero.

Transcript

00:01 This question we are given two conditions.

00:02 The first one is $2 ,200 for the hundred of shares and the cost of four 300 shares is $4 ,800.

00:13 And what we have to do is we have to assume the relation in between these two conditions that the cost and share is linear and express cost as a function of cheer.

00:24 So what we are asked to do is if or is cost and x is cheers then we have to find the function y of x so this is what we are asked to do in this question so oh so as we assume that the relation in between these two conditions is linear so i'm drawing drawing straight line in between them and we have two conditions so the p2 is the second condition that is 300 years have cost 4800 dollars and the first condition is 100 years and the first condition is 100 years and the cost is $2 ,200 so these are two conditions and in order to find the our function we have to find the slope of that particular equation so for the slope we use this particular relation y2 minus y1 by x2 minus x1 and now we block the values from these two numbers so 4800 minus 22 hundred 300 minus hundred it will be 26 hundred divide by 200 so it will be 13 so the slope of equation is 13 and we have the equation of line that is y minus y1 equals to m x minus x1 so to get the equation of this particular linear function of the eastern line now we have the slope of that particular equation yeah over here and we can plug either of one of the these two numbers or the point and plug it over here and we'll get the equation of this line.

03:11 So let's take the point one that is 2200 for 100 years.

03:19 Y minus 2200 and x minus 100 and as we have the slope as 13 so we distribute this equation we will get 13 x minus 1 ,300 and if we take this 2200 in the other side of equation we'll get plus 2200 over here and the equation will be 13 x plus 900 so this is the function y equals to f x it says 13x plus 1900 900.

04:23 So in this function, we are expressing x in terms of y in terms of x, which means we are expressing cost, that is y in terms of x, which means we are expressing cost, that is y, in terms of x, which is here...

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