SOLUTION: The total cost of production varied linearly with the number produced. When 20 were produced, the cost was $450. When 30 were produced, the cost was $650. Write an equation that gi

Question 1033141: The total cost of production varied linearly with the number produced. When 20 were produced, the cost was $450. When 30 were produced, the cost was $650. Write an equation that gives the cost as a function of the number produced, and find out what it would cost if 100 were produced.
--- I do not understand what it means to vary linearly. I understand varying inversely, and varying directly, but linearly is a new concept for me. An explanation of how to solve this word problem would be greatly appreciated. Thanks!
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
i believe they mean that the cost function will generate a straight line.
a linear equation has the maximum exponent of the variable x equal to 1.
y = x is a linear equation.
y = 5 is a linear equation.
y = x + 5 is a linear equation.
y = x^2 + 5 is not because the exponent on the variable is greater than 1.

a linear function will have a constant increase or decrease.

you are given that:
when 20 are produced, the cost was 450.
when 30 were produced, the cost was 650.

the slope intercept form of the equation for a straight line is y = mx + b.
m is the slope.
b is the y-intercept.

the slope is given by (y2-y1) / (x2-x1)

(x1,y1 is one point on the line.
(x2,y2) is another point on the line.

you are given 2 points.

they are (20,450) and (30,650)

if you let (x1,y1) = (20,450) and (x2,y2) = (30,650), then your slope will be (650 - 450) / (30 - 20) which is equal to 200 / 10 which can be simplifies to 20.

your slope is 20 so the equation becomes y = 20x + b

to find b, just replace x and y with the values from one of the points on the line.

using (20,450), the equation becomes 450 = 20*20 + b which becomes 450 = 400 + b.
subtract 400 from both sides of the equation to get 450 - 400 = b which results in b = 50.

your equation is now y = 20x + 50.

when x = 20, the equation tells you that y = 450.
when x = 30, the equation tells you that y = 650.

this agrees with what was given.

when x = 100, the equation tells you that y = 20*100 + 50 which becomes y = 2000 + 50 which becomes y = 2050.

when 100 are produced, the cost will be 2050.

the graph below shows the relationship and the value derived through the formula.

$$$

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