The base of a cone is fixed to the top of a cylinder to make a decoration - OCR - GCSE Maths - Question 14 - 2020 - Paper 6

The volume V of the cone can be given by the formula:

$V_{cone} = \frac{1}{3} \pi r^{2} h_{cone} = \frac{1}{3} \pi r^{2} (5) = \frac{5}{3} \pi r^{2}.$

The volume V of the cylinder is calculated using the formula:

$V_{cylinder} = \pi r^{2} h_{cylinder} = \pi r^{2} (1) = \pi r^{2}.$

The total volume of the decoration is the sum of the volumes of the cone and the cylinder:

$V_{total} = V_{cone} + V_{cylinder} = \frac{5}{3} \pi r^{2} + \pi r^{2} = 225.$

Rearranging the total volume equation gives:

$\frac{5}{3} \pi r^{2} + \pi r^{2} = 225.$\n We can factor out (\pi r^{2}):

$\pi r^{2}\left(\frac{5}{3} + 1\right) = 225.$

To combine the fractions:

$\frac{5}{3} + 1 = \frac{5}{3} + \frac{3}{3} = \frac{8}{3}.$\

So we have:

$\pi r^{2} \cdot \frac{8}{3} = 225.$\n

Solving for (r^{2}):

$r^{2} = \frac{225 \cdot 3}{8 \pi}.$\n

Calculating this gives:

$r^{2} = \frac{675}{8\pi}.$ \n

Taking the square root gives:

$r = \sqrt{\frac{675}{8\pi}}.$