Question

Mountains officials want to build a new ski lift from B to C, as shown in the figure below. The distance from A to C is 1450 feet. They measure angle DAC to be 37° and angle DBC to be 22°. What is the distance from A to B? Round your answer to the nearest tenth of a foot.

Answer 1

Let the height of the mountain be z

Hence the height of the mountain is:

`\begin{gathered} \sin37^(\circ)=(h)/(1450) \\ h=872.6 \end{gathered}`

From the figure it follows:

`\begin{gathered} \cos37^(\circ)=(DA)/(1450) \\ DA=1158.0 \end{gathered}`

Also from the figure it follows:

`\tan22^(\circ)=(h)/(DB)`

Reall h=872.6.

Substitute h=872.6 into the equation:

`\begin{gathered} \tan22^(\circ)=(872.6)/(DB) \\ DB=(872.6)/(\tan22^(\circ)) \\ DB=2159.8 \end{gathered}`

Using segment addition it follows:

`DB=DA+AB`

Substitute DA=1158.0 and DB=2159.8 into the equation:

`2159.8=1158+AB`

Solve for AB:

`\begin{gathered} AB=2159.8-1158 \\ AB=1001.8 \end{gathered}`

Therefore the distance from A to be is 1001.8 feet.