(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 i...
Question 1 Report
(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.
Answer Details
(a) Cylindrical pipe, length \(28\text{ m}=2800\text{ cm}\), internal radius \(r=3.5\text{ cm}\), external radius \(R=5\text{ cm}\).
(i) Volume of metal used.
The metal is the hollow shell between the outer and inner cylinders:
\[V_{\text{metal}}=\pi(R^2-r^2)\times L\]\[R^2-r^2=5^2-3.5^2=25-12.25=12.75\text{ cm}^2\]\[V_{\text{metal}}=\frac{22}{7}\times 12.75\times 2800\]\[=\frac{22}{7}\times 35700=22\times 5100=112200\text{ cm}^3\]The volume of metal is \(112200\text{ cm}^3\).
(ii) Volume of water the pipe holds when full.
This is the inner cylinder's volume:
\[V_{\text{water}}=\pi r^2 L=\frac{22}{7}\times 3.5^2\times 2800\]\[=\frac{22}{7}\times 12.25\times 2800=\frac{22}{7}\times 34300=22\times 4900=107800\text{ cm}^3\]Convert to litres using \(1\text{ litre}=1000\text{ cm}^3\):
\[\frac{107800}{1000}=107.8\text{ litres}\]The pipe holds \(107.8\) litres (to 1 d.p.).
(b) Tangent \(MP\) at \(M\), chord \(LN\parallel MP\): show \(\triangle LMN\) is isosceles.
By the tangent-chord (alternate segment) theorem, the angle between tangent \(MP\) and chord \(MN\) equals the angle in the alternate segment standing on \(MN\):
\[\angle PMN=\angle MLN \quad\text{...(1)}\]Since \(LN\parallel MP\) and \(MN\) is a transversal, alternate angles are equal:
\[\angle PMN=\angle MNL \quad\text{...(2)}\]From (1) and (2):
\[\angle MLN=\angle MNL\]In triangle \(LMN\) the base angles at \(L\) and \(N\) are equal, so the sides opposite them are equal, i.e. \(|MN|=|ML|\).
Therefore triangle \(LMN\) is isosceles. (Q.E.D.)
Download The App On Google Playstore
Everything you need to excel in your exams
