Distance Calculation for Cyclist Travel

Problem Analysis

The question asks for the total distance a cyclist covered. We are given the total time taken and the speeds for the first and second halves of the journey. The key is that the distance is divided into two equal parts.

Key Formulas

• Distance = Speed $\times$ Time
• Time = Distance / Speed

Step-by-Step Solution

Let the total distance be $D$ km.

• The first half of the distance is $\frac{D}{2}$ km.
• The second half of the distance is $\frac{D}{2}$ km.
• Speed for the first half ($v_1$) = 7 km/h.
• Speed for the second half ($v_2$) = 6 km/h.
• Total time taken ($T$) = 13 h.

Calculating Time Components

The time taken for the first half ($t_1$) is:

$t_1 = \frac{\text{Distance}}{\text{Speed}} = \frac{D/2}{v_1} = \frac{D}{2 \times 7} = \frac{D}{14}$ hours.

The time taken for the second half ($t_2$) is:

$t_2 = \frac{\text{Distance}}{\text{Speed}} = \frac{D/2}{v_2} = \frac{D}{2 \times 6} = \frac{D}{12}$ hours.

Finding the Total Distance

The total time is the sum of the times for the two halves:

$T = t_1 + t_2$

Substitute the known values:

$13 = \frac{D}{14} + \frac{D}{12}$

To solve for $D$, find a common denominator for 14 and 12, which is 84:

$13 = \frac{6 \times D}{84} + \frac{7 \times D}{84}$

Combine the terms on the right side:

$13 = \frac{6D + 7D}{84}$

$13 = \frac{13D}{84}$

Now, solve for $D$:

$D = \frac{13 \times 84}{13}$

$D = 84$ km.

The total distance covered by the cyclist is 84 km.