Question

TREES From point A,

the angle of elevation to the top of a pine tree is 42°.
From point B,
on the same side of the tree, the angle of elevation to the top of the tree is 50°.


If point A
and point B
are located 12
feet apart and are both at ground level, what is the approximate height of the tree to the nearest foot?

Answer 1

44 ft

Explanation:

To find the height of the tree, h, model the given scenario as a two right triangles and solve using the tan trigonometric ratio.

`\boxed{\begin{minipage}{7 cm}\underline{Tan 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}}`

See attached for a diagram modelling the given information.

Create two tan ratios using the given information, and rearrange each equation to isolate x.

Equation 1

Given:

• θ = 42°
• O = h
• A = x + 12

Therefore:

`\begin{aligned}\implies \tan 42^(\circ)&=(h)/(x+12)\\\\ x+12&=(h)/(\tan 42^(\circ))\\\\ x&=(h)/(\tan 42^(\circ))-12\end{aligned}`

Equation 2

Given:

• θ = 50°
• O = h
• A = x

`\begin{aligned}\implies \tan 50^(\circ)&=(h)/(x)\\\\ x&=(h)/(\tan 50^(\circ))\end{aligned}`

To find the value of h, substitute equation 2 into equation 1 to eliminate x and solve for h:

`\implies (h)/(\tan 50^(\circ))=(h)/(\tan 42^(\circ))-12`

`\implies (h)/(\tan 50^(\circ))-(h)/(\tan 42^(\circ))=-12`

`\implies (h\tan 42^(\circ)-h\tan 50^(\circ))/(\tan 50^(\circ)\tan 42^(\circ))=-12`

`\implies \left((\tan 42^(\circ)-\tan 50^(\circ))/(\tan 50^(\circ)\tan 42^(\circ))\right)h=-12`

`\implies h=(-12\tan 50^(\circ)\tan 42^(\circ))/(\tan 42^(\circ)-\tan 50^(\circ))`

`\implies h=44.1967977...`

Therefore, the approximate height of the tree to the nearest foot is 44 ft.