# Proving and using Sine Law? Question provided below...Sara is riding in a hot air balloon, directly above john, who is standing on the ground. Sara spots Pat on the ground at angle of depression of...

Proving and using Sine Law? Question provided below...

Sara is riding in a hot air balloon, directly above john, who is standing on the ground. Sara spots Pat on the ground at angle of depression of 28 degrees. The balloon rises 34m. Now the angle of depression is 35 degrees. How far is Pat from John?

Please help!

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Let the horizontal distance between John and Pat be x, the vertical distance between John and the balloon is y, and the hypotenuse be h.

==> h^2 = x^2 + y^2............(1).

Now, the balloon rises 34 m.

The distance between John and Pat is the same (x), the vertical distance is (y+34), and hypotenuse is h2.

==> h2^2 = x^2 + (y+34)^2

==> h2^2 = x^2 + y^2 + 68y + 1156

But x^2 + y^2 = h^2

==> h2^2 = h^2 +68y + 1156 ...............(2)

Now we will use the sine.

==> y= h sin(28)...........(3)

==> x = h cos(28).........(4)

==> x= h2 cos35...........(5)

==> Divide (4) by (5).

==> 1 = h/h2 (cos28/cos35)

==> h2 = (cos28/cos35)h .............(6)

Now we will substitute (6) and (3) in (2).

==> h2^2 = h^2 + 68y + 1156

==> (cos28/cos35)^2 h^2 = h^2 + 68(h*sin28) + 1156

==> 1.162 h^2 = h^2 + 31.924 h + 1156

==> 0.162 h^2 - 31.924 h - 1156 = 0

Now we will use the quadratic formula to find the roots.

==> h= (31.924 +42.05 ) / .324 = 228.32 m

==> x= h cos28 = 228.32 cos28 = 201.591 m

==> y= hsin28 = 228.32 sin28 = 107.19 m

**Then, the distance between Pat and John is x= 201.591 m**

Another method using tangent's and right angles

Angle of depression = 90 - angle.

Let x be the original height.

Let y be the distance between Pat and John

(1) y/x = tan(90-28)

(2) y/(x+34) = tan(90-35)

So from (1)

y=xtan(62)

Substituting into (2)

x tan(28)/(x+34)=tan(55)

x tan(62) = (x+34)tan(55)

x tan(62) = x tan(55) + 34 tan(55)

x (tan(62) - tan(55)) = 34 tan(55)

x = (34 tan(55))/(tan(62)-tan(55))

x = 107.29 m.

Distance between Pat and John would be

`y=107.29(tan(62^o))=201.78269m`