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`

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**

## We’ll help your grades soar

Start your 48-hour free trial and unlock all the summaries, Q&A, and analyses you need to get better grades now.

- 30,000+ book summaries
- 20% study tools discount
- Ad-free content
- PDF downloads
- 300,000+ answers
- 5-star customer support

Already a member? Log in here.

Are you a teacher? Sign up now