Gasoline sales at bob’s service station have shown a steady increase over the first 10 months of the year. which model relates the gasoline sold as a function of the number of months since the beginning of the year? how many gallons of gasoline will be sold in the months of november and december, and during which month will gasoline sales exceed 50,000 gallons?

[Incomplete question. I found a table in other source – see attachment]

I made a graph and it looks like a line (see attachment).

If the proportion is exactly linear, you have to use this kind of equations:

y=mx+b

where:

y are the sales

x is the number of months since the beginning of the year (for example, January is 1, February is 2, etc)

m is the proportion (how much it increases every month)

b is how much it was sold in month 0 (if January is 1, 0 would be December of last year)

But it’s not exactly linear – you have to round a little to make it straight. So now we need to know the equation of the line that fits best with these data. The most accurate way to know that is the method of least squares – no one actually calculates that, not even university professors: we just put data in Excel, click “add trendline”, and Excel does all the work ☺ Or you can just eyeball it and draw a line that looks right – in this case, make dY/dX to find m and the intersection with y-axis to find b.

This is what Excel told me:
y = 1.2727x + 30.54

So if you want to know how much will be sold in November, assuming the rate goes on like this, you have to replace 11 in the equation:

y = 1.2727*11 + 30.54=44.539

So that would mean  44,539 gallons

And 12 for Dezember:

y = 1.2727*12 + 30.54=45.812

So that would mean 45,812 gallons

So you will not exceed 50,000 until next year.

Let’s find in which month of next year:

1.2727x + 30.54 = 50

1.2727x = 50 - 30.54

x = (50 - 30.54)/1.2727=12.147

So this means it will be in month 13 (you have to round it up)

which would be January (you have to substract 12)

the mistake in the pattern is that :

b. the decimal point was moved one too many place values each time

the correct pattern could be made by :

b. moving the decimal point to the right the same numbers of place values as the exponents