A delivery truck driver records the odometer reading and the remaining fuel in the gas tan… — ACT

A delivery truck driver records the odometer reading and the remaining fuel in the gas tank at two points during a long-distance trip. At an odometer reading of $140$ miles, the gas tank contains $18$ gallons of fuel. At an odometer reading of $290$ miles, the tank contains $12$ gallons of fuel. Assuming the fuel is consumed at a constant rate, what is the slope of the line that models the amount of fuel remaining (in gallons) as a function of the odometer reading (in miles)?

$-\frac{1}{25}$ Answer
B
$-25$
C
$\frac{1}{25}$
D
$25$
E
$-\frac{1}{15}$

Answer

The slope of the line modeling remaining fuel relative to odometer distance is

-\frac{1}{25}

gallons per mile.

The slope of a line passing through

(x_1, y_1)

and

(x_2, y_2)

is given by

m = \frac{y_2 - y_1}{x_2 - x_1}

. Assigning

x

to miles and

y

to gallons gives points

(140, 18)

and

(290, 12)

. Substituting these coordinates yields

\frac{12 - 18}{290 - 140} = \frac{-6}{150} = -\frac{1}{25}

. This negative value represents a rate of decrease of

1

gallon for every

25

miles driven.

Step-by-Step Solution

Identify the given coordinate points

(x, y)

where

x

represents the odometer reading in miles and

y

represents the remaining fuel in gallons.

The two points are

(140, 18)

and

(290, 12)

The problem specifies fuel remaining as a function of odometer reading, making distance the independent variable (

x

) and fuel the dependent variable (

y

Apply the slope formula

m = \frac{y_2 - y_1}{x_2 - x_1}

m = \frac{12 - 18}{290 - 140} = \frac{-6}{150}

Slope measures the vertical change (change in fuel) divided by the horizontal change (change in distance).

Simplify the resulting fraction

\frac{-6}{150}

m = -\frac{1}{25}

Dividing both the numerator and denominator by their greatest common divisor,

6

, yields

-\frac{1}{25}

Key Concept

Slope as Rate of Change

Estimated Time:1m 15s

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