Answer:
132 feet
Step-by-step explanation:
Given that Alejandro is standing at a distance of 140 feet from the base of the tree and his eyes are 6 feet above the ground.
Let AB is the height of the tree and point E is the location of his eyes which is 6 feet above from C on the ground as shown in the figure.
The distance between points A and C, AC=140 feet.
Drawing a horizontal line from point E which meets AB at point D as shown.
As ACED forms a rectangle, so
AC=DE=140 feet ...(i)
CE=AD= 6 feet ...(ii)
The height of the tree, AB=AD+DB
By using equation (ii), AB=6+DB ...(iii)
Now, given that the on watching the top of the tree, the reading on the clinometer is 42 degrees.
So,[tex]\angle DEB = 42^{\circ}[/tex]
In triangle DEB,
[tex]\tan 42^{\circ} = \frac {DB}{DE} \\\\\Rightarrow DB = DE \times \tan 42^{\circ} \\\\[/tex]
[tex]\Rightarrow DB = 140 \times \tan 42^{\circ}[/tex] [from (i)]
[tex]\Rightarrow DB = 126[/tex] feet
From equation (iii) the height of the tree is
AB=6+126=132 feet.
Hence, the height of the tree is 132 feet.
