The diagram shows a cylinder and a cone - OCR - GCSE Maths - Question 14 - 2018 - Paper 1

The volume of a cone is given by the formula:

$V_{cone} = \frac{1}{3} \pi r^2 h$

We need to express the height of the cone in terms of r. Given the ratio r : h = 1 : 4, we can say:

$h = 4r$\n\nNow substituting this in the volume formula for the cone:

$V_{cone} = \frac{1}{3} \pi r^2 (4r) = \frac{4}{3} \pi r^3$.

Since the volume of the cone is equal to the volume of the cylinder:

$\frac{4}{3} \pi r^3 = 36\pi$\n\nWe can then cancel (\pi) from both sides:

$\frac{4}{3} r^3 = 36$.

To solve for r, first multiply both sides by 3:

$4r^3 = 108$\n Next, divide both sides by 4:

$r^3 = 27$\n Now take the cube root:

$r = 3$ cm.

Thus, the value of r is (3) cm.