The diagram shows a circle, centre O - Edexcel - GCSE Maths - Question 23 - 2019 - Paper 3

The equation of the circle centered at O can be expressed as:

$x^2 + y^2 = r^2$

Substituting the known coordinates for point B (16, 0) into the equation gives us:

$16^2 + 0^2 = r^2 \implies r^2 = 256$

Thus, the radius is 16.

Now, substituting point P's coordinates (3p, p) into the equation of the circle:

$(3p)^2 + p^2 = 256$

Expanding and simplifying gives:

$9p^2 + p^2 = 256 \implies 10p^2 = 256 \implies p^2 = 25.6$

Therefore, taking the square root:

$p = \sqrt{25.6}$. After calculating, we find the value of p.

Calculating the square root yields:

$p \approx 5.06$

Rounding this to 1 decimal place, we conclude that:

$p = 5.1$