Question

The angle of elevation to the top of a building in New York is found to be 15 degrees from the ground at a distance of 1 mile from the base of the building. Using this information, find the height of the building. (Round your answer to four decimal places.)

Answer 1

`0.2679`

Explanation:

Consider the given figure.

Let AB denotes the height of a building.

BC = 1 mile

To find: AB

`\tan C` = side opposite to the angle/side adjacent to the angle

`\tan C=(AB)/(BC)`

Here,
`C=15^(\circ)\,,\,BC=1\,\,mile`

So,

`\tan 15^(\circ)=(AB)/(1)\\\Rightarrow AB=\tan 15^(\circ)`

Here,

`\tan 15^(\circ)=\tan \left (45^(\circ)-30^(\circ) \right )\\=(\tan 45^(\circ)-\tan 30^(\circ))/(1+\tan 45^(\circ)\tan 30^(\circ))\,\,\left \{ \because \tan (A-B)=(\tan A-\tan B)/(1+\tan A\tan B) \right \}\\=(1-(1)/(√(3)))/(1+(1)/(√(3)))\\=(√(3)-1)/(√(3)+1)\\=(1.73205-1)/(1.73205+1)\\=(0.73205)/(2.73205)\\=0.2679`

Therefore,

`AB=0.2679`