Prove that the height of the cloud , Mathematics

HEIGHTS AND DISTANCES

If the angle of elevation of cloud from a point 'h' meters above a lake is α and the angle of depression of its reflection in the lake is  β, prove that the height of the cloud is.

1206_height and distance.png

Ans : If the angle of elevation of cloud from a point 'n' meters above a lake is ∝ and the angle of depression of its reflection in the lake is β, prove that the height of the cloud is h

( tan β + tan α / tan β - tan α )

Let AB be the surface of the lake and

Let p be an point of observation such that AP = h meters. Let c be the position of the cloud and c' be its reflection in the lake. Then ∠CPM = ∝ and ∠ MPC1 = β.

Let CM = x.

Then, CB = CM + MB = CM + PA = x + h

In ? CPM, we have tan ∝ = CM /PM

⇒    tan ∝ =  x/AB

[∴ PM = AB]

⇒ AB = x cot∝                        ...........1

In ? PMC', we have

tanβ = C 'M/PM

⇒  tanβ =  x + 2h/AB [Θ C'M=C'B+BM = x + h + n]

⇒  AB = (x + 2h) cot β

From 1 & 2

x cot ∝ = (x + 2h) cot β

x (cot ∝ - cot β) = 2h cot β (on equating the values of AB)

x= (  1/tan α -1/tan β) = 2h /tan β = x(tan β - tan α/tan α - tan β) = 2h /tan β

x = 2h  tan α /tan β - tan α

Hence, the height CB of the cloud is given by CB is given by CB = x + h

⇒ CB= 2h  tan α /tan β - tan α + h

2h tan α    + h tan β - tan α

⇒ CB- 2h tan α + h tan β - h tan α /tan β - tanα= h (tan α + h tan β )/tan β - tan α

Posted Date: 4/9/2013 2:13:36 AM | Location : United States







Related Discussions:- Prove that the height of the cloud , Assignment Help, Ask Question on Prove that the height of the cloud , Get Answer, Expert's Help, Prove that the height of the cloud Discussions

Write discussion on Prove that the height of the cloud
Your posts are moderated
Related Questions
If an instrument has precision of +-1, can it detect a value of 1.3?

If the lengths of all sides of a box are doubled, how much is the volume increased? a. 2 times b. 4 times c. 6 times d. 8 times d. The volume of a box is taken by mu

The longer base of a trapezoid is three times the shorter base. The nonparallel sides are congruent. The nonparallel side is 5 cm more that the shorter base. The perimeter of the t

Find the are of the rectilinear.if it is the difference between to isosceles trapezoid whose corrsponding sides are parallel.

A car buyer has a choice of three makes, five body styles, and six colors. How many different choices does the buyer have?

An experiment designed to test the potency of a drug on 20 rats. Last animal studies have shown that a 10 mg dose of the drug is lethal 5% of the time within the first 4 hours; of

Suppose a Ferris wheel with radius of 12 meters is rotating at a rate of 2 rotations per minute. a. How fast is a person rising when the person is 3 meters above the horizontal lin


Product Moment Coefficient (r) This gives an indication of the strength of the linear relationship among two variables.                                     N

OTHER WAYS TO AID LEARNING :  Here we shall pay particular attention to the need for repetition, learning from other children, and utilising errors for learning.