A particle moves along the parabola `y=x^(2)` in the first quadrant in such a way that its x-coordinate (measured in metres) increases at a rate of 10 m/sec. If the angle of inclination `theta` of the line joining the particle to the origin change, when x = 3 m, at the rate of k rad/sec., then the value of k is

Options

A. 1
B. 2
C. `1//2`
D. `1//3`

Explanation

To solve the problem step-by-step, we will analyze the motion of the particle along the parabola \(y = x^2\) and find the rate of change of the angle of inclination \(\theta\) of the line joining the particle to the origin when \(x = 3\) m.

### Step 1: Understand the relationship between \(x\), \(y\), and \(\theta\)

The particle moves along the curve defined by \(y = x^2\). The coordinates of the particle are \((x, y) = (x, x^2)\). The angle \(\theta\) is defined as the angle of inclination of the line joining the particle to the origin \((0, 0)\).

### Step 2: Express \(\tan(\theta)\)

The slope of the line from the origin to the point \((x, y)\) is given by:
\[ \tan(\theta) = \frac{y}{x} = \frac{x^2}{x} = x \]
Thus, we have:
\[ \theta = \tan^{-1}(x) \]

### Step 3: Differentiate \(\tan(\theta)\) with respect to time \(t\)

To find how \(\theta\) changes with time, we differentiate both sides with respect to \(t\):
\[ \frac{d}{dt}(\tan(\theta)) = \frac{d}{dt}(x) \]
Using the chain rule:
\[ \sec^2(\theta) \frac{d\theta}{dt} = \frac{dx}{dt} \]

### Step 4: Substitute known values

We know that \(\frac{dx}{dt} = 10\) m/s (the rate at which \(x\) is increasing). We need to find \(\sec^2(\theta)\) when \(x = 3\).

### Step 5: Calculate \(\theta\) when \(x = 3\)

At \(x = 3\):
\[ y = x^2 = 3^2 = 9 \]
Now we can find \(\tan(\theta)\):
\[ \tan(\theta) = \frac{y}{x} = \frac{9}{3} = 3 \]
Thus:
\[ \theta = \tan^{-1}(3) \]

### Step 6: Calculate \(\sec^2(\theta)\)

Using the identity:
\[ \sec^2(\theta) = 1 + \tan^2(\theta) \]
we have:
\[ \sec^2(\theta) = 1 + 3^2 = 1 + 9 = 10 \]

### Step 7: Substitute back into the differentiated equation

Now substituting \(\sec^2(\theta)\) into the differentiated equation:
\[ 10 \frac{d\theta}{dt} = 10 \]
Dividing both sides by 10 gives:
\[ \frac{d\theta}{dt} = 1 \text{ rad/sec} \]

### Conclusion

Thus, the rate of change of the angle of inclination \(\theta\) when \(x = 3\) m is:
\[ k = 1 \text{ rad/sec} \]