A small sphere has a radius of 1.5 cm - Junior Cycle Mathematics - Question 14 - 2015

Let the radius of the large sphere be ( R ). The volume of the large sphere is three times that of the small sphere:

$V_{large} = 3 V_{small} = 3 \left( \frac{9}{2} \pi \right) = \frac{27}{2} \pi$

Using the volume formula for the large sphere:

$V_{large} = \frac{4}{3} \pi R^3$

Setting the two expressions for volume equal:

$\frac{4}{3} \pi R^3 = \frac{27}{2} \pi$

Dividing both sides by ( \pi ):

$\frac{4}{3} R^3 = \frac{27}{2}$

Multiplying both sides by ( \frac{3}{4} ):

$R^3 = \frac{27 \times 3}{2 \times 4} = \frac{81}{8}$

Taking the cube root:

$R = \sqrt[3]{\frac{81}{8}} = \frac{\sqrt{81}}{\sqrt[3]{8}} = \frac{9}{2} \text{ cm}$

Thus, the radius of the large sphere is ( \frac{9}{2} \text{ cm} ).