A manufacturer produces a storage tank - Edexcel - A-Level Maths Pure - Question 1 - 2019 - Paper 2

To find the surface area of the tank, we need to combine the surface areas of the cylindrical and hemispherical parts.

1. Surface Area of Cylinder: The lateral surface area of the cylinder is given by: $A_c = 2 imes ext{π} imes r imes h$

2. Surface Area of Hemisphere: The surface area of the hemisphere is: $A_h = 2 imes ext{π} imes r^2$

3. Total Surface Area: Thus, the total surface area A of the tank is: $A = A_c + A_h = 2 imes ext{π} imes r imes h + 2 imes ext{π} imes r^2$

4. Volume Constraint: Since the volume V of the tank is given by: $V = ext{π} r^2 h + \frac{2}{3} ext{π} r^3$ and it equals 6 m³, we can rearrange to express h in terms of r: $h = \frac{6 - \frac{2}{3} \text{π} r^3}{\text{π} r^2}$

5. Substituting into Surface Area: Substitute h back into the total surface area equation to express it solely in terms of r: $A = 2 \text{π} r \left( \frac{6 - \frac{2}{3} \text{π} r^3}{\text{π} r^2} \right) + 2 \text{π} r^2$ After simplifying, we arrive at: $A = \frac{12}{r} + \frac{5}{3} \text{π} r^2$