Determine the height of both platforms.

Question

Tatiana is standing on a platform ready to make a dive. She notices that the angle of depression to her coach sitting at the end of the pool is 12°. On her next dive, she walks up vertically 6m to the higher

platform and notices that the angle of depression to her coach is now 17°. Determine the height of

both platforms.

Answer 1

the horizontal distance "d" to the coach is the same in both cases so tan(12°) = h₁/d and tan(17°) = h₂/d

But h₂ = h₁ + 6 so:

tan(12°) = h₁/d and tan(17°) = (h₁+6)/d

or (from 1st equation): d = h₁/tan(12°) and (from the 2nd equation) d = (h₁ + 6)/tan(17°)

Since both are equal to "d", we can set them equal to each other:

h₁/tan(12°) = (h₁ + 6)/tan(17°)

Now you can cross multiply to get:

h₁•tan(17°) = (h₁ + 6)•tan(12°)

h₁•tan(17°) = h₁•tan(12°) + 6tan(12°)

h₁•tan(17°) - h₁•tan(12°) = 6tan(12°)

h₁[tan(17°) - tan(12°)] = 6tan(12°)

h1 = 6tan(12°)[tan(17°) - tan(12°)]

h₁ = 13.69 m

so h₂ = 19.69 m

Answer 2

Check the diagram in the following link:

https://www.autodraw.com/share/SXDX2RMIW8Z9

Let d be the horizontal distance from the platform to the end of the pool where the coach is sitting.

Let h be the vertical height of the lower platform (with angle of depression =12)

and h+6 the vertical height of the upper platform (with angle of depression = 17)

Look at the two right triangles formed.

Remember Tan(angle)=(opposite leg)/(adjacent leg)

Tan(12)= h/d

Tan(17)=(h+6)/d

tan 12/tan 17=(h/d)/((h+6)/d) (when you write the ratio of tan 12 over tan 17, the d's cancel each other.)

tan 12/tan 17=h/(h+6)

tan12/tan 17≈ 0.7

so, we have h/(h+6) = 0.7

solve for h by cross multiplication h=0.7(h+6) ==> h=14 will be the approximate height of the lower platform and 14+6=20 will be the approximate height of the upper platform.