A manufacturer produces a storage tank - Edexcel - A-Level Maths Pure - Question 14 - 2019 - Paper 2

To find the surface area of the tank, we need to consider both the cylindrical and hemispherical parts. The surface area of the cylinder (excluding the closed end) is given by:

$A_{cylinder} = 2 \pi r h$

The surface area of the hemisphere is given by:

$A_{hemisphere} = 2 \pi r^2$

Thus, the total surface area A is:

$A = A_{cylinder} + A_{hemisphere} = 2 \pi r h + 2 \pi r^2$

Given the volume V of the tank is:

$V = \pi r^2 h + \frac{2}{3} \pi r^3 = 6$

We can express h in terms of r:

$h = \frac{6 - \frac{2}{3} \pi r^3}{\pi r^2}$

Substituting h back into the surface area equation gives:

$A = 2 \pi r \left( \frac{6 - \frac{2}{3} \pi r^3}{\pi r^2} \right) + 2 \pi r^2$

Simplifying this expression leads to:

$A = \frac{12}{r} + \frac{5}{3} \pi r^2$.