A man on a cliff observes a boat at an angle 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. Find the total time taken by the boat to reach the shore.
Answers
Answer:
Explanation:
Hi ,
Draw a rough diagram to the data
given in the question,
|A
|
|
|
| h
|
| 60
|________|_____30
B <----y----->C.<-x--> D
Join A to C and D
Height of the cliff = AB = h
BC = y
Distance travelled in 6 mi
= CD = x
i ) in traingle ADB ,
Angle ABD = 90
Tan 30 = h / ( x + y )
1/ √3 = h / ( x + y )
( x + y )/( √3 ) = h -----( 1 )
ii ) In triangle ABC
Angle B = 90
Tan 60 = h / y
√3 = h / y
( √ 3 ) y = h ------( 2 )
From ( 1 ) and ( 2 )
We observe that
( 1 ) = ( 2 )
( x + y ) / (√ 3 ) = ( √3 ) y
x + y = 3y
x = 3y - y
x = 2y -----( 3 )
iii ) according to the problem given
distance ' x ' is travelled in
6 minutes
How much time taken to
travel distance ( x + y ) = ?
Time taken
= [ 6 × ( x + y ) ] / x
= [ 6 × ( 2y + y ) ] / 2y
{ since x = 2y from ( 3 ) ]
= [ 6 × 3y ] / 2y
= 18 y / 2y
= 9 minutes
Therefore ,
In 9 minuates the boat
reaches the shores .
I hope this helps you. :)