Problem Statement
A metallic hollow cylindrical pipe of length $14\text{ cm}$ has an outer diameter of $10\text{ cm}$ and an inner diameter of $6\text{ cm}$. Find the volume of metal contained in the pipe and the total surface area of the pipe. (Take $\pi = 22/7$)
Verified Solution & Marking Scheme
Extract Pipe Radii and Height
Outer radius $R = 10/2 = 5\text{ cm}$, Inner radius $r = 6/2 = 3\text{ cm}$, Length $h = 14\text{ cm}$.
Calculate Volume of Metal
$\text{Volume} = \pi (R^2 - r^2) h = \frac{22}{7} (5^2 - 3^2) \times 14 = 22 \times 2 \times (25 - 9) = 44 \times 16 = 704\text{ cm}^3$
Calculate Total Surface Area (TSA)
$\text{TSA} = \text{External CSA} + \text{Internal CSA} + 2(\text{Area of Circular Base Rings})$
$= 2\pi R h + 2\pi r h + 2\pi(R^2 - r^2) = 2\pi [h(R + r) + (R^2 - r^2)]$
$= 2 \times \frac{22}{7} [14(5 + 3) + (25 - 9)] = \frac{44}{7} [14(8) + 16] = \frac{44}{7} [112 + 16] = \frac{44}{7} \times 128 = \frac{5632}{7} \approx 804.57\text{ cm}^2$