The angle from a stake in the ground to the top of a tree is 60 degrees. If the height of the tree is 40 feet, what is the distance from the stake to the bottom of the tree (the distance along the gound) in feet? Please explain your answer!

Here we are given that the angle from stake in ground to the top of the tree is 60° and the height of the tree is 40ft . We are interested in finding out the distance from stake to the bottom of the tree . For figure refer to the attachment .

From the figure , BC = 40ft which is the perpendicular of the right angled triangle ABC . Also assume that AB = x ft , which is the base of the triangle .

Now since we have base and perpendicular , we should use the ratio of tangent as ,

[tex]\rm\longrightarrow tan\theta =\dfrac{perpendicular}{base}=\dfrac{p}{b} [/tex]

And here [tex]\theta[/tex] = 60° . On substituting the respective values , we have ;

[tex]\rm\longrightarrow tan 60^o =\dfrac{40\ ft}{x \ }\\ [/tex]

And the value of tan60° = √3 , so that ;

[tex]\rm\longrightarrow \sqrt3 =\dfrac{40\ ft}{x} [/tex]

Cross multiply ,

[tex]\rm\longrightarrow x =\dfrac{40}{\sqrt3}ft. [/tex]

The value of √3 is 1.732 approximately .

[tex]\rm\longrightarrow x =\dfrac{40}{1.732}ft. [/tex]

Simplify,

[tex]\rm\longrightarrow \underline{\underline{\red{{x =23.09\ ft }}}} [/tex]

And we are done !