Math Problem Statement

A freight train is carrying goods across the country. The number of gallons of fuel it has used varies directly with the distance it has traveled. See the graph below.

Solution

To solve this, we’ll examine the graph and address each question.

Observations from the Graph

The graph shows a linear relationship between the distance traveled (in miles) on the x-axis and the number of gallons used on the y-axis.

  1. At approximately 100 miles, the train has used 100 gallons.
  2. At approximately 400 miles, the train has used 400 gallons.

Since this is a direct variation, we can calculate the fuel usage rate and the slope.

Solution for Part (a): Gallons Used per Mile

The gallons per mile is determined by the ratio of gallons to miles. Here’s how we can find it:

Rate=Change in gallonsChange in miles=400100400100=300300=1 gallon per mile\text{Rate} = \frac{\text{Change in gallons}}{\text{Change in miles}} = \frac{400 - 100}{400 - 100} = \frac{300}{300} = 1 \text{ gallon per mile}

So, the train uses 1 gallon per mile.

Solution for Part (b): Slope of the Graph

The slope of a line in a graph showing direct variation is equivalent to the rate calculated above, as the slope represents "gallons per mile."

Therefore, the slope is also 1.

Summary of Answers

  • (a) The train uses 1 gallon per mile.
  • (b) The slope of the graph is 1.

Would you like further details on this solution or have any questions?

Related Questions

  1. How would the slope change if the train used more gallons per mile?
  2. What would the graph look like if the fuel usage rate was 0.5 gallons per mile?
  3. How can we interpret the y-intercept in the context of this problem?
  4. What does it mean if the slope were greater than 1?
  5. How would the graph change if the train carried more weight?

Tip:

In direct variation problems, identifying two clear points can simplify finding the rate or slope.

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Math Problem Analysis

Mathematical Concepts

Direct Variation
Linear Functions
Slope Calculation

Formulas

Slope formula: (y2 - y1) / (x2 - x1)
Direct Variation formula: y = kx

Theorems

Direct Variation

Suitable Grade Level

Grades 6-8