A variable `DeltaABC` is inscribed in a circle of diameter `x` units. At a particular instant the rate of change of side `a`, is `x/2` times the rate of change of the opposite angle `A` then `A =`

Options

A. `pi//6`
B. `pi//3`
C. `pi//4`
D. `pi//2`

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the relationship between the side and angle in a triangle inscribed in a circle.
The side \( a \) of triangle \( \Delta ABC \) can be related to the angle \( A \) using the sine rule. For a triangle inscribed in a circle, we have:

\[ \frac{a}{\sin A} = 2R \]

where \( R \) is the radius of the circumscribed circle. Given that the diameter of the circle is \( x \), the radius \( R \) is:

\[ R = \frac{x}{2} \]

### Step 2: Substitute the radius into the sine rule.
Substituting \( R \) into the sine rule gives:

\[ \frac{a}{\sin A} = x \]

From this, we can express \( a \) in terms of \( A \):

\[ a = x \sin A \]

### Step 3: Differentiate both sides with respect to \( A \).
Now, we differentiate both sides with respect to \( A \):

\[ \frac{da}{dA} = x \cos A \]

### Step 4: Relate the rates of change.
According to the problem, the rate of change of side \( a \) is \( \frac{x}{2} \) times the rate of change of angle \( A \):

\[ \frac{da}{dA} = \frac{x}{2} \cdot \frac{dA}{dA} \]

This simplifies to:

\[ \frac{da}{dA} = \frac{x}{2} \]

### Step 5: Set the two expressions for \( \frac{da}{dA} \) equal.
Now we have two expressions for \( \frac{da}{dA} \):

\[ x \cos A = \frac{x}{2} \]

### Step 6: Solve for \( \cos A \).
Dividing both sides by \( x \) (assuming \( x \neq 0 \)):

\[ \cos A = \frac{1}{2} \]

### Step 7: Find the angle \( A \).
The angle \( A \) for which \( \cos A = \frac{1}{2} \) is:

\[ A = \frac{\pi}{3} \text{ radians} \quad \text{(or 60 degrees)} \]

Thus, the value of angle \( A \) is:

\[ \boxed{\frac{\pi}{3}} \]

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