Question

Need help explain to me

Answer 1

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Answer:

  A) (20 miles)/(1 gallon) = 20 miles per gallon

  B) 20/1 = 20

Step-by-step explanation:

After you read the problem statement and identify the questions being asked, it is a good idea to look carefully at the given information. Graphs A and B are identical in every way.

The vertical axis is labeled "y" and "Distance (miles)". The horizontal axis is labeled "x" and "Gallons". The graph starts at (0 gallons, 0 miles) and goes through points on the grid that are 2 squares up for each 1 square to the right. Each square up represents 10 miles. Each square to the right represents 1 gallon.

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A) Pick two points on the graph. It is usually convenient to choose points where the graph crosses grid intersections. It is often convenient to choose one of them as (0, 0) if that point is on the graph (it is). I expect it to be convenient to choose the second point as (1 gallon, 20 miles).

The change in distance between the point (0 gallons, 0 miles) and (1 gallon, 20 miles) is (20 miles) - (0 miles) = 20 miles.

The change in gallons between the point (0 gallons, 0 miles) and (1 gallon, 20 miles) is (1 gallon) - (0 gallons) = 1 gallon.

The ratio of these changes is (20 miles)/(1 gallon) = 20 miles per gallon.

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B) We can use the same two points. Here, we basically ignore the units.

Change in y = 20-0 = 20.

Change in x = 1 -0 = 1.

Slope = 20/1 = 20.

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Additional comment

For many problems, I find it convenient to keep the units with the numbers. Making sure the units work out properly is called "units analysis." Doing that can save you from a number of mistakes where units are involved.

Units can be treated just like a variable--multiplied, divided, raised to a power, cancelled from fractions. And, like with variables, terms can only be added or subtracted if they have the same units.

Here, we have ...

  $\dfrac{\text{change in distance}}{\text{change in gallons}}=\dfrac{20\text{ miles}}{1\text{ gallon}}=\dfrac{20}{1}\times\dfrac{\text{miles}}{\text{gallon}}=20\text{ miles per gallon}$

Note that "divided by" and "per" mean essentially the same thing in this context.

As you can see from the work done with Graph A, the slope of the line has units that are the ratio of the units of the y-axis to the units of the x-axis. (miles/gallon) Even though in Graph B, we write the slope as 20 without any units, it will only make any sense to use that number where units of miles/gallon make sense.