To find out how far the dog has to run to catch up to the cat, we need to find the distance at which they are equal.
Let's analyze the situation step by step:
1. Distance covered by the dog = x
2. Distance covered by the cat = (x + 12) since they start 12 meters apart.
According to the given information, for every 7 meters that the dog runs, the cat runs 4 meters. So we can set up a proportion to compare their distances:
Distance covered by the dog / Distance covered by the cat = 7 / 4
Substituting the variables, we get:
x / (x + 12) = 7 / 4
To solve for x, we can cross-multiply:
4x = 7(x + 12)
Now, we can solve for x:
4x = 7x + 84
Subtracting 7x from both sides:
-3x = 84
Dividing by -3:
x = -28
However, since distance cannot be negative, we discard this solution.
From the equation, we can see that the dog will never catch up to the cat since the ratio of their speeds is fixed (7:4) and the cat starts 12 meters ahead. The dog would have to run indefinitely to catch up to the cat, but will never actually reach it.