Question

Question #77163

Answer 1

Malia is `62.4m` from the spire.

Explanation:

Let's first break the diagram up into two right-angled triangles and name the vertices. Working anticlockwise from the top of the spire:

`S`= top of the spire. (assume it is vertical)
`L` = Leah's position on the left.
`B` = the base of the spire
`M` = Malia's position on the right.

You have to have 3 facts to use trigonometry in a triangle:

• A `90°` angle,
• one side,
• another side or an angle.

Malia's triangle only has two; the angle of elevation and `90°`

In `Delta SLB` we can find the height of the spire which is common to both triangles.

`(SB (opp))/(120 (adj)) = tan18°` which gives `" "color(blue)(SB = 120tan18°)`

You could calculate the answer, but you would have to round off. Let's leave it like this for the meantime.

In `Delta SMB` we now have `3` facts because we know the height of the spire, `SB`.

`(SB ( opp))/(x ( adj)) = tan32°" "larr` invert to get `x` on top

`x/(SB) = 1/(tan32°)" "larr` isolate `x`

`x = color(blue)(SB)/ (tan32°)" "larr` use the previous answer for `SB`

`x = color(blue)(120tan18°)/(tan32°)`

`x = 62.398m`

Hope this helps?

Answer 2

Malia is standing 62.39 meters from the base of the spire.

Explanation:

First we want to find `y`, or the height of the spire. For this we can use the tangent of the angle, represented by:

`tan theta=("opposite side"("spire"))/"adjacent side"`

so:

`"Spire"=tan theta xx "adjacent"`
`S=tan 18 xx 120`
`S = 0.3249 xx 120`
`S=38.99" m"`

We can do the same with Malia's position except that now we want to know what is the length of the adjacent side:

`tan theta="Spire"/"Adjacent"" "` so, `" ""Adjacent"= "Spire"/tan theta`
`A=S/tan 32`

`A=38.99/0.6248`

`A=62.39" m"`