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

Given that the volume of the cone is equal to the volume of the cylinder, we have:

$V_{cone} = V_{cylinder}$

From the cone's volume formula:

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

Setting these equal, we have:

$\frac{1}{3} \pi r^{2} h = 36\pi$

From the ratio given, ( r : h = 1 : 4 ), we can express h as:

$h = 4r$

Substituting this into the volume equation gives:

$\frac{1}{3} \pi r^{2} (4r) = 36\pi$

Simplifying the equation:

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

Dividing both sides by (\pi):

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

Multiplying through by 3:

$4r^{3} = 108$

Now divide by 4:

$r^{3} = 27$

Taking the cube root of both sides yields:

$r = 3 \text{ cm}$