Answer:
3.2 m
Step-by-step explanation:
Let x be the distance between the cliff and the boat.
Since the cliff is 25 m above sea level and the distance between the boat and the height of the cliff are perpendicular, and the line of sight of the boat to the base of a cliff is 29 , these three form a right-angled triangle with hypotenuse side as the line of sight of the elevation of the boat to the base of the cliff.
So, using trigonometric ratios, tan29° = 25/x
x = 25 m/tan29° = 25 m/0.5543 = 45.1 m
Also, the line of sight of the top of the flagpole, the cliff and the distance between the boat and the cliff form a right-angled triangle with hypotenuse side as the line of sight of the top of the flagpole,
Now, the angle of elevation of the top of the flagpole from the top of the boat is 32°, we now find the angle opposite to x, the distance from the boat to the cliff.
Since it is a right-angled triangle, thus one angle is 90° and the unknown angle is A.
So, 90° + 32° + A = 180° (sum of angles in a triangle)
122° + A = 180°
A = 180° - 122°
A = 58°
Let y be the length of the flagpole. In this same triangle, using the sine rule, with x being the side opposite A and 25 + y being the side opposite the 32° angle, where 25 + y = height of cliff + length of flagpole, then
x/sinA = (25 + y)/sin32°
25 + y = xsin32°/sinA
y = xsin32°/sinA - 25
y = 45.1sin32°/sin58° - 25
y = 45.1 × 0.5299/0.8480 - 25
y = 45.1 × 0.6249 - 25
y =28.183 - 25
y = 3.183
y ≅ 3.2 m to one decimal place
exists.