9. (a) A buoy in the form of a hollow spherical shell of external radius 0.7 m and internal radius 0.65 m floats in water - Leaving Cert Applied Maths - Question 9 - 2018

To find the percentage of the volume of the buoy immersed, we need to calculate the buoyant force and equate it to the weight of the buoy.

1. Volume of the Buoy (V):

   The volume of the hollow sphere is given by:

   $V = \frac{4}{3} \pi (R^3 - r^3)$ Where:

   • $R = 0.7 \, \text{m}$ (external radius)
   • $r = 0.65 \, \text{m}$ (internal radius)

   Hence:

   $V = \frac{4}{3} \pi (0.7^3 - 0.65^3)$

2. Weight of the Buoy (W):

   The weight can be calculated using the density of the shell:

   $W = \text{density} \times V = 3430 \times V$

3. Buoyant Force (B):

   The buoyant force is given by the formula:

   $B = \text{Volume submerged} \times \text{Density of water} \times g$

Since the buoy is floating:

$B = W$

From the above, we can derive:

$1000 \times \left(\frac{1}{3} \pi (0.7)^2 h_s \right) = 3430 \times \left(\frac{4}{3} \pi (0.7)^3 - \frac{4}{3} \pi (0.65)^3 \right)$

4. Percentage Immersed:

   Finally, the percentage of the volume immersed:

   $\text{Percentage} = \left( \frac{V_s}{V} \right) \times 100$ where $V_s$ is the submerged volume.