Dog Catch Cat Chase Calculation

This problem involves calculating the time taken for the dog to catch the cat, considering their speeds and the initial distance between them. This is a classic relative speed problem.

Understanding Relative Speed

When two objects move in the same direction, the speed at which the faster object gains on the slower object is called the relative speed. It is calculated as the difference between their speeds.

• Dog's speed ($v_d$) = 24 km/h
• Cat's speed ($v_c$) = 10 km/h
• Relative speed ($v_{rel}$) = $v_d - v_c$

Calculating the relative speed:

$v_{rel} = 24 \text{ km/h} - 10 \text{ km/h} = 14 \text{ km/h}$

Calculating Catch Up Time

The dog needs to cover the initial distance separating it from the cat at this relative speed.

• Initial distance ($d$) = 280 m

First, ensure the units are consistent. Convert the distance to kilometers:

$d = 280 \text{ m} = \frac{280}{1000} \text{ km} = 0.28 \text{ km}$

Now, use the formula: Time = Distance / Speed

Time ($t$) = $\frac{d}{v_{rel}} = \frac{0.28 \text{ km}}{14 \text{ km/h}}$

$t = 0.02 \text{ hours}$

Converting Time to Minutes

The options are given in minutes, so convert the calculated time from hours to minutes.

1 hour = 60 minutes

$t \text{ (in minutes)} = 0.02 \text{ hours} \times 60 \text{ minutes/hour}$

$t = 1.2 \text{ minutes}$

Therefore, the dog will take 1.2 minutes to catch the cat.