Two cyclists start from the junction of two perpendicular roads, there velocities being `3um//m in` and `4um//m in` , respectively. Find the rate at which the two cyclists separate.

Answer

To solve the problem, we need to find the rate at which the two cyclists are separating after they start from a junction of two perpendicular roads. Let's break down the solution step by step.

### Step 1: Understand the Problem
We have two cyclists starting from the same point (the junction) and moving in perpendicular directions. One cyclist travels at a speed of \(3u\) m/min, and the other at \(4u\) m/min. We want to find the rate at which the distance between them is increasing.

### Step 2: Define Variables
Let:
- \(t\) = time in minutes
- Distance covered by the first cyclist after time \(t\) = \(d_1 = 3u \cdot t\)
- Distance covered by the second cyclist after time \(t\) = \(d_2 = 4u \cdot t\)

### Step 3: Use Pythagorean Theorem
Since the cyclists are moving in perpendicular directions, we can use the Pythagorean theorem to find the distance \(x\) between them:
\[ x^2 = d_1^2 + d_2^2 \]
Substituting the distances:
\[ x^2 = (3u \cdot t)^2 + (4u \cdot t)^2 \]
\[ x^2 = 9u^2 t^2 + 16u^2 t^2 \]
\[ x^2 = 25u^2 t^2 \]
Taking the square root:
\[ x = 5u \cdot t \]

### Step 4: Differentiate with Respect to Time
To find the rate at which the distance \(x\) is increasing, we differentiate \(x\) with respect to \(t\):
\[ \frac{dx}{dt} = \frac{d}{dt}(5u \cdot t) \]
\[ \frac{dx}{dt} = 5u \]

### Step 5: Conclusion
The rate at which the two cyclists are separating is \(5u\) m/min.