Math  /  Algebra

QuestionTwo trains leave the station at the same time, one heading west and the other east. The westbound train travels 16 miles per hour slower than the eastbound train. If the two trains are 540 miles apart after 3 hours, what is the rate of the westbound train?
Do not do any rounding. \square 7 miles \$er hour

Studdy Solution

STEP 1

1. Two trains leave the station at the same time, one heading west and the other east.
2. The westbound train travels 16 miles per hour slower than the eastbound train.
3. After 3 hours, the two trains are 540 miles apart.
4. We need to find the rate of the westbound train.

STEP 2

1. Define variables for the rates of the eastbound and westbound trains.
2. Write an equation that models the total distance covered by both trains.
3. Solve the equation for the rate of the westbound train.

STEP 3

Define variables for the rates of the eastbound and westbound trains.
Let r r be the rate of the eastbound train in miles per hour. The rate of the westbound train is r16 r - 16 miles per hour.

STEP 4

Write an equation that models the total distance covered by both trains.
The distance covered by the eastbound train in 3 hours is 3r 3r . The distance covered by the westbound train in 3 hours is 3(r16) 3(r - 16) .
The total distance between the two trains after 3 hours is 540 miles, so:
3r+3(r16)=540 3r + 3(r - 16) = 540

STEP 5

Solve the equation for the rate of the westbound train.
First, simplify the equation:
3r+3r48=540 3r + 3r - 48 = 540
Combine like terms:
6r48=540 6r - 48 = 540
Add 48 to both sides to isolate the term with r r :
6r=588 6r = 588
Divide both sides by 6 to solve for r r :
r=5886 r = \frac{588}{6} r=98 r = 98
The rate of the eastbound train is 98 miles per hour.
The rate of the westbound train is:
r16=9816=82 r - 16 = 98 - 16 = 82
The rate of the westbound train is:
82 \boxed{82}

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