Distance Calculation: Delhi to Agra Cars

This problem involves calculating the distance covered by a car based on relative speeds and travel times.

• Total Distance (Delhi to Agra): 233 km
• Speed of Car P ($v_P$) = Speed of Car Q ($v_Q$) + 10 km/hr
• Car Q starts 1 hour after Car P.
• Cars cross 75 minutes after Car Q starts.

Time Analysis for Cars P and Q

We need to determine the duration each car travelled until they met.

• Time Car Q travelled ($t_Q$) = 75 minutes. Convert to hours: $t_Q = \frac{75}{60} \text{ hours} = 1.25 \text{ hours}$.
• Car P started 1 hour earlier than Car Q. So, the time Car P travelled ($t_P$) is: $t_P = t_Q + 1 \text{ hour} = 1.25 + 1 = 2.25 \text{ hours}$.

Speed Calculation using Distance Equation

When the cars cross, the sum of the distances they travelled equals the total distance between Delhi and Agra.

Distance = Speed × Time

Distance covered by P ($d_P$) = $v_P \times t_P$

Distance covered by Q ($d_Q$) = $v_Q \times t_Q$

The equation is: $d_P + d_Q = 233$ km.

Substitute the known values and the relationship between speeds:

$(v_Q + 10) \times 2.25 + v_Q \times 1.25 = 233$

Now, solve for $v_Q$:

$2.25 v_Q + (10 \times 2.25) + 1.25 v_Q = 233$

$2.25 v_Q + 22.5 + 1.25 v_Q = 233$

Combine the $v_Q$ terms:

$(2.25 + 1.25) v_Q + 22.5 = 233$

$3.5 v_Q = 233 - 22.5$

$3.5 v_Q = 210.5$

$v_Q = \frac{210.5}{3.5}$

$v_Q = \frac{2105}{35} = \frac{421}{7}$ km/hr.

Distance Travelled by Car Q

To find how many kilometers Car Q travelled, use its speed ($v_Q$) and the time it travelled ($t_Q$).

Distance Q ($d_Q$) = $v_Q \times t_Q$

$d_Q = \frac{421}{7} \times 1.25$

$d_Q = \frac{421}{7} \times \frac{5}{4}$

$d_Q = \frac{2105}{28}$ km.

Calculating the value:

$d_Q \approx 75.17857$ km.

Rounding to one decimal place, the distance travelled by Car Q is approximately 75.2 km.