(a) A cylindrical pipe is 28 metres long. Its internal radius is 3.5 cm and external radius 5 cm. Calaulate : (i) the volume, in cm(^{3}), of metal used in making the pipe ; (ii) the volume of water in litres that the pipe can hold when full, correct to 1 decimal place. [Take (pi = frac{22}{7})]
(b) In the diagram, MP is a tangent to the circle LMN at M. If the chord LN is parallel to MP, show that the triangle LMN is isosceles.
Explanation
(a)(i) Volume of outer pipe = (pi r^{2} h)
= (frac{22}{7} times (5.0)^{2} times 28 times 100)
Volume of inner pipe = (frac{22}{7} times (3.5)^{2} times 28 times 100)
Volume of metal used = Volume of outer pipe – Volume of inner pipe
= (22 times 400 times (5.0^{2} – 3.5^{2}))
= (22 times 400 times (5.0 – 3.5) times (5.0 + 3.5))
= (112,200 cm^{3})
(ii) Vol of water the pipe can hold = vol of inner pipe
= (frac{22}{7} times (3.5)^{2} times 28 times 100)
= (107,800 cm^{3})
= 107.8 litres.
(b) < PMN = < LNM (alternate angles)
< PMN = < LMN (angles in alternate segment)
(therefore) < LNM = < NLM, then (Delta) LMN is isosceles.