Question

the mean distance of two points taken at random on the circumference of a circle with radius r is given by
a
1. 4r/π
2. r/π
3. 3r/π
4. 2r/π ​

Answer 1

Answer: Mean distance of two points taken at random on the circumference of a circle with radius r is $\frac{4r}{\pi}$.

Given: A circle of radius r.

To Find: Mean distance of two points taken at random on the circumference of a circle with radius r.

Step-by-step explanation:

Step 1:

We may apply the law of cosines to calculate the distance between the points, which results in:

$(distance)^2 = r^2 +r^2-2(r)(r)cos\theta\\(distance)^2 = 2r^2-2 r^2 cos\theta\\distance = \sqrt (2r^2- 2 r^2 cos\theta)$

Step 2: Using a clever little trick of the sine half angle identity, we can reduce this expression.:

$sin(x/2) = \sqrt((1-cos \ x)/2)$

so then

$2sin(x/2) = \sqrt(2-2cos \ x)$

Since the value of sine is positive for θ in [0, π], we can write:

$distance = |2 sin(\theta/2)| = 2 sin(\theta/2)$

Step 3: The average value of the above function needs to be determined. From the definition of the mean of a function over an interval, this calculation is simple.

Step 4: The integral of the function from a to b multiplied by 1/(b - a) is the mean of a function f(x) throughout the range [a, b]. Since the angle for the distance function ranges between [0, π], the average value is:

$=\frac{1}{\pi -0}\int\limits^\pi_0 {2rsin\frac{\theta}{2} } d\theta\\\\=\frac{1}{\pi} (-4r \ cos \frac{\theta}{2})\limits^\pi_0\\\\=\frac{4r}{\pi}$

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