This problem involves calculating the speed of runners in a marathon race given the distance, the difference in their finishing times, and the difference in their speeds. We can use the fundamental relationship between distance, speed, and time: Time = Distance / Speed.

Analyzing the Runner Speed Problem

Let's break down the information given in the problem:

• Total distance of the marathon race: 20 km
• Difference in finishing times: 30 minutes
• Difference in speeds: 2 km/h

We need to find the speed of the winner, who is the faster runner.

Setting Up Equations for Runner Speeds

Let $v_w$ be the speed of the winner (in km/h) and $v_l$ be the speed of the slower runner (in km/h).

The problem states that their speeds differ by 2 km/h. Since the winner is faster, we have:

\begin{equation*} v_w - v_l = 2 \end{equation*}

From this, we can express the slower runner's speed in terms of the winner's speed:

\begin{equation*} v_l = v_w - 2 \end{equation*}

Now, let's consider the time taken by each runner. The time taken to cover the 20 km distance is:

• Time taken by the winner ($t_w$): $t_w = \frac{\text{Distance}}{\text{Winner's Speed}} = \frac{20}{v_w}$ hours
• Time taken by the slower runner ($t_l$): $t_l = \frac{\text{Distance}}{\text{Slower Runner's Speed}} = \frac{20}{v_l}$ hours

We know that the time difference is 30 minutes. We need to convert this to hours:

\begin{equation*} 30 \text{ minutes} = \frac{30}{60} \text{ hours} = 0.5 \text{ hours} \end{equation*}

Since the winner finishes 30 minutes earlier, the slower runner's time is 30 minutes (or 0.5 hours) more than the winner's time:

\begin{equation*} t_l - t_w = 0.5 \end{equation*}

Substitute the expressions for $t_w$ and $t_l$ into this equation:

\begin{equation*} \frac{20}{v_l} - \frac{20}{v_w} = 0.5 \end{equation*}

Now, substitute $v_l = v_w - 2$ into the equation:

\begin{equation*} \frac{20}{v_w - 2} - \frac{20}{v_w} = 0.5 \end{equation*}

Solving for the Winner's Speed

We now have an equation with only one variable, $v_w$. Let's solve for $v_w$:

Find a common denominator on the left side:

\begin{equation*} \frac{20v_w - 20(v_w - 2)}{v_w(v_w - 2)} = 0.5 \end{equation*}

Simplify the numerator:

\begin{equation*} \frac{20v_w - 20v_w + 40}{v_w^2 - 2v_w} = 0.5 \end{equation*}

\begin{equation*} \frac{40}{v_w^2 - 2v_w} = 0.5 \end{equation*}

Multiply both sides by $(v_w^2 - 2v_w)$:

\begin{equation*} 40 = 0.5 (v_w^2 - 2v_w) \end{equation*}

Distribute 0.5:

\begin{equation*} 40 = 0.5v_w^2 - v_w \end{equation*}

To get rid of the decimal, multiply the entire equation by 2:

\begin{equation*} 80 = v_w^2 - 2v_w \end{equation*}

Rearrange the equation into a standard quadratic form $av^2 + bv + c = 0$:

\begin{equation*} v_w^2 - 2v_w - 80 = 0 \end{equation*}

We can solve this quadratic equation by factoring. We look for two numbers that multiply to -80 and add up to -2. These numbers are -10 and 8.

So, we can factor the quadratic equation as:

\begin{equation*} (v_w - 10)(v_w + 8) = 0 \end{equation*}

This gives us two possible solutions for $v_w$:

\begin{equation*} v_w - 10 = 0 \implies v_w = 10 \end{equation*}

or

\begin{equation*} v_w + 8 = 0 \implies v_w = -8 \end{equation*}

Since speed must be a positive value, the only valid solution for the winner's speed is $v_w = 10$ km/h.

Verification of the Runner Speeds

Let's check if this speed fits the conditions of the problem:

• Winner's speed $v_w = 10$ km/h.
• Slower runner's speed $v_l = v_w - 2 = 10 - 2 = 8$ km/h.
• Winner's time $t_w = \frac{20}{10} = 2$ hours.
• Slower runner's time $t_l = \frac{20}{8} = 2.5$ hours.

The difference in their times is $t_l - t_w = 2.5 - 2 = 0.5$ hours, which is equal to 30 minutes. This matches the problem statement perfectly.

Therefore, the winner's speed is 10 km/h.

Revision Table: Marathon Runner Problem

Additional Information: Distance, Speed, and Time Calculations

The relationship between distance, speed, and time is fundamental in physics and mathematics problems involving motion. The formulas are:

• Distance = Speed × Time
• Speed = Distance / Time
• Time = Distance / Speed

When dealing with problems involving two entities (like runners, cars, trains) with different speeds or times, it's often helpful to set up equations for each entity's time or distance and then relate them using the given differences or sums. Quadratic equations frequently arise in these types of problems when substituting variables leads to a squared term, as seen in the marathon runner calculation.

Remember to ensure all units (distance, speed, time) are consistent before performing calculations. In this problem, distance was in km, speed in km/h, so time needed to be in hours. The 30 minutes was converted to 0.5 hours.