Question

A flagstaff, 7.32 m long is fitted at the top of 10 m tall building. The flagstaff is supported by the ropes which are tied to the point P on the ground which is x m away from the base of the building. It is given that `l_1` is the length of rope from point P to the base of the flagstaff and `l_2` is the length of rope from point P to the top of flagstaff. Rope `l_1` makes an angle of 30° with the horizontal and θ be the angle which rope `l_2` makes with the horizontal as shown in the figure.

Based on the given information, answer the following questions:

 (Use `sqrt(2) = 1.4` and `sqrt(3) = 1.732`)

(i) Find the value of x.   [1]

(ii) Find the measure of angle θ.   [1]

(iii) (a) Find the total length of ropes needed to support the flagstaff.   [2]

OR

(iii) (b) Which rope is longer `l_1` or `l_2` and by how much?   [2]

Answer 1

(i) In ΔBCP,

`tan 30^circ = (BC)/(CP)`

`1/sqrt(3) = 10/x`

∴ `x = 10sqrt(3)`

x = 10 × 1.732

x = 17.32 m

(ii) In ΔACP,

`tan θ = (AC)/(CP)`

= `(AB + BC)/x`

= `(7.32 + 10)/(17.32)`

tan θ = 1

∴ θ = 45°

(iii) (a) `sin 30^circ = 10/(l_1)`

`1/2 = 10/(l_1)`

`l_1` = 20 m

`l_2 = sqrt((17.32)^2 + (17.32)^2)`

= `17.32sqrt(2)`

= 17.32 × 1.4

`l_2` = 24.248 m

 Required length `l_1 + l_2` = 20 m + 24.248 m

= 44.248 m

= 44.25 m

OR

(iii) (b) ∴ `l_2 - l_1` = 24.25 – 20

= 4.25 m

∴ `l_2` is larger by 4.25 m