A solid cone is joined to a solid hemisphere to make the solid T shown below - Edexcel - GCSE Maths - Question 18 - 2022 - Paper 2

To find the value of $y$, we first determine the volume of the solid T, which is comprised of a cone and a hemisphere:

1. Calculate the radius of the hemisphere and cone:
   The diameter of both the cone and hemisphere is $7$ cm, so the radius $r$ is:
   $r = \frac{7}{2} = 3.5 \text{ cm}$

2. Calculate the volume of the hemisphere (V_h):
   The volume of a hemisphere is given by the formula: $V_h = \frac{2}{3} \pi r^3$ Substituting the radius:
   $V_h = \frac{2}{3} \pi (3.5)^3 = \frac{2}{3} \pi \cdot 42.875 \approx 89.797 \text{ cm}^3$

3. Determine the volume of the cone (V_c):
   The volume of a cone is given by the formula: $V_c = \frac{1}{3} \pi r^2 h$
   Here, we do not yet know the height $h$, but from the total volume of T: $120 = V_h + V_c$ $V_c = 120 - V_h = 120 - 89.797 \approx 30.203 \text{ cm}^3$
   Therefore: $30.203 = \frac{1}{3} \pi (3.5)^2 h$ Simplifying this gives: $30.203 = \frac{1}{3} \pi \cdot 12.25 h$ $h = \frac{30.203 \cdot 3}{\pi \cdot 12.25} \approx 2.37 \text{ cm}$

4. Calculate the total height y of T:
   The total height of T consists of the height of the cone and the radius of the hemisphere: $y = h + r = 2.37 + 3.5 = 5.87 ext{ cm}$

5. Round to 3 significant figures:
   Thus, the value of $y$ is: $y \approx 5.87 ext{ cm}$