A man on cliff observes a boat at angle of of depression of 30° which is approaching the shore to the point immediately beneath the observer with a uniform speed six minutes later. The angle of depression of the boat is found to be 60°. if the the height of the cliff is 20cm then the distance between observer and the point p ?
Answers
Answer:
It is given in the problem that the time taken by the boat to the reach from the point A to the point D is 6 min and the speed of the boat is vm/min, then the distance covered from the point A to the point D is given as:
Distance=Speed×Time
Distance AD=6v
Now, assume that the boat takes t time to reach the shore then the distance covered from the point D to the point B is given as:
DB=vt
Now, apply the trigonometric ratio in the triangle DBC,
tan60∘=PerpendicularBase
We know that for the triangle DBC, BC is the perpendicular and DB is the base whose lengths are BC=handDB=vt, then we have
tan60∘=BCDB
⇒3–√=hvt
⇒h=vt3–√ … (1)
Now, apply the trigonometric ratio in the triangle ABC,
tan30∘=PerpendicularBase
We know that for the triangle ABC, BC is the perpendicular and AB is the base whose lengths are BC=h and AB=6v+vt, then we have
tan30∘=BCAB
13–√=hv(6+t)
⇒h=v(6+t)3–√ … (1)
Compare the values h from the equation (1) and equation (2),
vt3–√=v(6+t)3–√
Simplify the equation:
t3–√×3–√=6+t
Solve the equation for the value of t.
3t=6+t
⇒3t−t=6
⇒2t=6
⇒t=3minutes
Therefore, the boat will take 3 minutes to reach the shore.
As given that take 6 minute to reach the point D from the point A and we have find that the boat take 3 minute to reach the shore from the point D, os the total time taken is:
Total time=6+3=9 minute.
Step-by-step explanation:
