Question

From a point x = 80 feet in front of a public library, the angles of elevation to the base of the flagpole and the top of the flagpole are = 29.5° and 39° 45', respectively. The flagpole is mounted on the front of the library's roof. Find the height of the flagpole.

Answer 1

Let's draw the scenario to better understand the details.

To be able to determine the height of the flagpole, let's create two different triangles with 29.5° and 39° 45' angle. The two triangles have one common base at 80 Feet, yet have different heights at H+h and H respectively.

Where,

H = Height of the library

h = Height of the flag

The two triangles are proportional at a common base, thus, let's generate this expression using the Law of Sines:

`(H+h)/(\sin(39\degree45^(\prime)))\text{ = }(H)/(\sin(29.5^(\circ)))`

Let's simplify,

`(H+h)/(\sin(39\degree45^(\prime)))\text{ = }(H)/(\sin(29.5^(\circ)))\text{ }\rightarrow\text{ (}H+h)(\sin (29.5^(\circ)))\text{ = (H)(}\sin (39\degree45^(\prime)))`
`H\sin (29.5^(\circ))\text{ + h}\sin (29.5^(\circ))\text{ = H}\sin (39\degree45^(\prime))\text{ ; but }29.5^(\circ)=29^(\circ)30^(\prime)`
`H\sin (29^(\circ)30^(\prime))\text{ + h}\sin (29^(\circ)30^(\prime))\text{ = H}\sin (39\degree45^(\prime))`
`\text{h}\sin (29^(\circ)30^(\prime))\text{ = H}\sin (39\degree45^(\prime))\text{ - }H\sin (29^(\circ)30^(\prime))`
`\text{ h(}0.4924235601)\text{ = H(0.63943900198) -H}(0.4924235601)`
`\text{ h(}0.4924235601)\text{ = H(0.14701544188)}`