Hello!
The answer is:
The correct option is:
B) 83 feet
Why?
To solve the problem, we need to use the following trigonometric identity:
[tex]Tan(\alpha)=\frac{Opposite}{Adjacent}[/tex]
Which, translated to our problem, will be:
[tex]Tan(\alpha)=\frac{Height}{Base}[/tex]
We are given two triangles, and we know their height and their angles between their hypothenuse and their bases.
So,
For the first triangle, we have:
[tex]height=60ft\\\alpha=64\°[/tex]
So, using the trigometric identity of the tangent, we have:
[tex]Tan(\alpha)=\frac{Height}{Base}\\\\Tan(64\°)=\frac{60ft}{Base}\\\\Base=\frac{60ft}{Tan(64\°)}=\frac{60ft}{2.05}=29.27ft[/tex]
Therefore, we have that the base of the first triangle is equal to 29.27 feet.
For the second triangle, we have:
[tex]height=60ft\\\alpha=48\°[/tex]
So, using the trigometric identity of the tangent, we have:
[tex]Tan(\alpha)=\frac{Height}{Base}\\\\Tan(48\°)=\frac{60ft}{Base}\\\\Base=\frac{60ft}{Tan(48\°)}=\frac{60ft}{1.11}=54.05ft[/tex]
Therefore, we have that the base of the second triangle is equal to 54.05 feet.
Now, to calculate the horizontal distance between the two trees (x), we need to use the following formula with the obtained values of both triangles:
[tex]HorizontalDistance=FirstTriangleBase+SecondTriangleBase\\\\HorizontalDistance=29.27ft+54.05=83.29ft[/tex]
Hence, we have that the distance between the two trees rounded to the nearest foot is 83 feet.
Have a nice day!
