Question

The angle at the vertex of a cone is measured using a 30mm diameter circle. If the circle lies 3.72mm below the top of the cone, determine the value of angle θ.

A surveyor measured the angle of elevation of a flat spire as from a point on horizontal ground. He moves 30m nearer to the flat and measures the angle of elevation as . Calculate the height of the spire to the nearest hundredth. 

Answer 1

Let the cone be imagined as shown in the above figure.

Here AB is the diameter of the coin and NB is the radius of the coin.

From the question we have to find $\angle AMB$

Let $\angle NMB = \theta$

In $\triangle MNB$ right angled at N, $\sin{\theta}$ = $NB \div BM$

Using Pythagoras theorem $\lparen MN\rparen ^2 + \lparen BN\rparen ^2 = \lparen BM\rparen ^2$

= $\lparen 3.72\rparen ^2 + \lparen 15\rparen ^2 = \lparen BM\rparen ^2$

= 13.84 + 225

= 238.84

$\therefore BM = \sqrt{238.84}$

= 15.45

So $\sin{\theta} = 15\div15.45$

= 0.97

$\therefore \theta = \sin^{-1}\lparen0.97\rparen$

= 75.9$\degree$

$\therefore$ angle at the vertex = $2\theta$

= 2*75.9

= 151.8$\degree$

tan$\theta$ = $\frac{h-a}{d}$

d = 30 m

h = height of spire

$\theta$ = angle of elevation

a = height of surveyor

tan$\theta$ = $\frac{h-a}{30}$

30×tan$\theta$ = h - a

h = (30 × tan$\theta$ ) + a