Question

The graph shows the cost of producing gadgets. which equation best represents the cost C of producing n gadgets?

Answer 1

`C = 0.10n`

Explanation:

If you study the graph you will see that there is no cost for producing zero gadgets. Cost = $0

For 100 gadgets the cost is $10

For 200 gadgets the cost is $20

etc

So the cost in $ is 1/10 times the number of gadgets

For 1 gadget, it costs
`\$(1)/(10)` or $0.10

For n gadgets it should cost $
`0.1n` same as
`\$(n)/(10)`

Mathematically speaking the graph represents the equation of a straight line which is of the form

y = mx + b

where y represents the vertical axis and x the horizontal axis

The variable m represents the slope of the line which is the increase of y for every unit increase in x.

The slope, m can be determined by using the formula
`\displaystyle (y2-y1)/(x2-x1)` where x1, y1 and x2, y2 are two points on the graph

This is also called rise/run

b is called the y-intercept and it represents the value of y where the line crosses the vertical axis

The difference here is that instead of y, we are using C as the variable for the vertical(y) axis and n as the variable for the horizontal(x) axis

So we get the modified equation as
C = m·n + b

We can see that when n = 0, C = 0 so b = 0 because the vertical intercept is zero

To calculate m, take any two points on the line

I choose (0, 0) and (100,10)

`\rm Slope \;m = (10-0)/(100-0) = (10)/(100) = (1)/(10) = 0.10\\\\`

This means that for every gadget produced the cost will increase by $0.10

The equation becomes C = 0.10n + b and since b is 0 we get

`C = 0.10n`