Question

The angle of elevation to the top of a building changes from 15​° to ​30° as an observer advances 140 feet toward the building. Find the height of the​ building, x, to the nearest foot.

pls explain

Answer 1

70 ft

Explanation:

To find the height of the building, we can use the trigonometric relationship between the angle of elevation, the distance from the object, and the height of the object.

In this case, we have a right triangle formed by the observer, the top of the building, and the base of the building. Therefore, we can use the tangent trigonometric ratio, since the height of the building is the opposite the angle of elevation, and the distance between the observer and the building is the side adjacent the angle.

`\boxed{\begin{minipage}{7 cm}\underline{Tangent trigonometric ratio} \\\\$\sf \tan(\theta)=(O)/(A)$\\\\where:\\ \phantom{ww}$\bullet$ $\theta$ is the angle. \\ \phantom{ww}$\bullet$ $\sf O$ is the side opposite the angle. \\\phantom{ww}$\bullet$ $\sf A$ is the side adjacent the angle.\\\end{minipage}}`

Let "x" be the height of the building.

Let "d" be the initial distance from the observer to the building.

The angle of elevation changes from 15° to 30° as the observer advances 140 feet toward the building.

(See the attachment for a visual representation).

Based on this information, we can set up the following equations:

`\tan 15^(\circ)=(x)/(d)`

`\tan 30^(\circ)=(x)/(d-140)`

Rearrange both equations to isolate d:

`d=(x)/(\tan 15^(\circ))`

`d=(x)/(\tan 30^(\circ))+140`

Solve this system of equations by the method of substitution.

`(x)/(\tan 15^(\circ))=(x)/(\tan 30^(\circ))+140`

`(x)/(\tan 15^(\circ))-(x)/(\tan 30^(\circ))=140`

`(x\tan 30^(\circ)-x \tan 15^(\circ))/(\tan 30^(\circ)\tan 15^(\circ))=140`

`(x(\tan 30^(\circ)- \tan 15^(\circ)))/(\tan 30^(\circ)\tan 15^(\circ))=140`

`x=(140\tan 30^(\circ)\tan 15^(\circ))/(\tan 30^(\circ)- \tan 15^(\circ))`

`x=(140\cdot (√(3))/(3)(2-√(3)))/((√(3))/(3)- (2-√(3)))`

`x=((140(2√(3)-3))/(3))/((4√(3)-6)/(3))`

`x=(140(2√(3)-3))/(2(2√(3)-3))`

`x=(140)/(2)`

`x=70`

Therefore, the height of the building is exactly 70 feet.