When driving on a highway, Ian passed the 23 mile marker with 15 gallons of gas in the tank. When he passed the 73 mile marker, he had 12.5 gallons of gas left.

Question

A. If the number of gallons left in the tank is the dependent variable, find the slope of a graphed line that passes through these two points. What would the slope represent in terms of the context?

B. How much gas would Ian have left when passing the 100 mile marker? Round to the nearest gallons. State what strategy and tool you will use to answer the question, explain your choice, and then find the answer.

C. How would the graph and rate of change be different if the gas left were the independent variable and the distance traveled were the dependent variable?

Answer 1

You drove 50 miles with 2.5 gallons. That's 50/2.5 = 20 miles per gallon

23 to 73 is 50 miles. 15-12.5 = 2.5 gallons

(23,15) and (73,12.5) are two points on the graph of a line with slope = (15-12.5)/(23-73) = -2.5/50 =-1/20 That slope is the negative inverse of miles per gallon. It's the slope of a perpendicular line.

at the 100 mile marker, you've gone another 100-73 = 27 miles. At 20 mpg, you've used 27/20 = 1.35 gallons. 12.5-1.35 = 11.15 gallons left = 11 gal. rounded off to nearest gallon.

The graph of the line through the two points (23,15) and (73,12.5) is a downward sloping straight line, but close to flat, just slightly downward sloping.

It's equation is y=(-1/20)x + b, solve for b by plugging in either point

15=(-1/20)23 + b

b = 15+23/20 = 16 3/20 = 16.15

y=(-1/20)x + 16.15 or

y=-0.05x + 16.15

if you change dependent to independent variables, that's switching x and y

x = -0.05y + 16.15

then solve for y

-0.05y = x-16.15

y =(-1/0.05)x + 16.15/.05

y =-20x + 323

has rate of change that's much more negative, it's the inverse of the slope of y= (-1/20)x +16.15

y=-20x +323

23 =-20(15) +323 = 323-300

73 =-20(12.5) + 323 = 323-250

100=-20(11.15) + 323 = 323 - 223