Math Problem Statement

Suppose you have entered a 131-mile biathlon that consists of a run and a bicycle race. During your run, your average velocity is 5 miles per hour, and during your bicycle race, your average velocity is 29 miles per hour. You finish the race in 7 hours. What is try distance of the run? What is the distance of the bicycle race?

Solution

We can solve this problem using the concept of time and velocity. Let’s define the variables and then use the relationship between distance, velocity, and time:

  • Let the distance of the run be drd_r miles.
  • Let the distance of the bicycle race be dbd_b miles.

Step 1: Establish Equations

We know the following:

  • Total distance of the biathlon is 131 miles: dr+db=131d_r + d_b = 131
  • Average velocity during the run is 5 miles per hour, so the time spent running is dr5\frac{d_r}{5} hours.
  • Average velocity during the bicycle race is 29 miles per hour, so the time spent cycling is db29\frac{d_b}{29} hours.
  • Total time for both events is 7 hours: dr5+db29=7\frac{d_r}{5} + \frac{d_b}{29} = 7

Step 2: Solve the System of Equations

We have the system of two equations:

  1. dr+db=131d_r + d_b = 131
  2. dr5+db29=7\frac{d_r}{5} + \frac{d_b}{29} = 7

From equation (1), solve for dbd_b:

db=131drd_b = 131 - d_r

Substitute this into equation (2):

dr5+131dr29=7\frac{d_r}{5} + \frac{131 - d_r}{29} = 7

Now, solve for drd_r.

Step 3: Solve for drd_r

To eliminate the fractions, multiply through by the least common denominator of 5 and 29, which is 145: 145(dr5)+145(131dr29)=145×7145 \left( \frac{d_r}{5} \right) + 145 \left( \frac{131 - d_r}{29} \right) = 145 \times 7

This simplifies to: 29dr+5(131dr)=101529d_r + 5(131 - d_r) = 1015

Distribute the 5 on the left: 29dr+6555dr=101529d_r + 655 - 5d_r = 1015

Combine like terms: 24dr+655=101524d_r + 655 = 1015

Subtract 655 from both sides: 24dr=36024d_r = 360

Solve for drd_r: dr=36024=15d_r = \frac{360}{24} = 15

Step 4: Solve for dbd_b

Now substitute dr=15d_r = 15 into equation (1): db=13115=116d_b = 131 - 15 = 116

Final Answer:

  • The distance of the run is 15\mathbf{15} miles.
  • The distance of the bicycle race is 116\mathbf{116} miles.

Would you like further details or clarification?


Here are 5 related questions to consider:

  1. How would the solution change if the total time was 8 hours instead of 7?
  2. What would happen if the running speed increased to 6 miles per hour?
  3. How would you solve the problem if the total distance of the biathlon changed to 150 miles?
  4. Can you derive the formula for distance using average velocity and time?
  5. How long would the run take if the distance was increased by 5 miles?

Tip: When solving systems of equations, always try to simplify one variable before substitution. This minimizes errors.

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Systems of Equations
Distance-Time Relationship

Formulas

Distance = Velocity × Time
System of equations: d_r + d_b = 131 and d_r/5 + d_b/29 = 7

Theorems

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Suitable Grade Level

Grades 8-10