Question

Caleb and Emily are standing 100 yards from each other. Caleb looks up at a 45° angle to see a hot air balloon. Emily looks up at a 60° angle to see the same hot air balloon. Approximately how far is the hot air balloon off the ground?

Answer 1

B

Explanation:

Answer 2

63.39 yards.

Explanation:

Refer the attached figure

We are given that Caleb and Emily are standing 100 yards from each other i.e. BC = 100

Let BD = x

So, DC = 100-x

We are given that Caleb looks up at a 45° angle to see a hot air balloon i.e. ∠ABD = 45° and Emily looks up at a 60° angle to see the same hot air balloon i.e. ∠ACD = 60°

Let AD be the height of the balloon denoted by h.

In ΔABD

Using trigonometric ratio

`tan \theta = (Perpendicular)/(Base)`

`tan 45^(\circ) = (AD)/(BD)`

`1= (h)/(x)`

`x=h` ---1

In ΔACD

Using trigonometric ratio

`tan \theta = (Perpendicular)/(Base)`

`tan 60^(\circ) = (AD)/(DC)`

`√(3)= (h)/(100-x)`

`√(3)(100-x)=h` ---2

So, equating 1 and 2

`√(3)(100-x)=x`

`100√(3)-√(3)x=x`

`100√(3)=x+√(3)x`

`100√(3)=x(1+√(3))`

`(100√(3))/(1+√(3))=x`

`63.39=x`

Thus the height of the balloon is 63.39 yards.