Answer:
The ballonist is at a height of 3579.91 ft above the ground at 3:30pm.
Step-by-step explanation:
Let's call:
h the height of the ballonist above the ground,
a the distance between the two observers,
[tex]a_1[/tex] the horizontal distance between the first observer and the ballonist
[tex]a_2[/tex] the horizontal distance between the second observer and the ballonist
[tex]\alpha _1[/tex] and [tex]\alpha _2[/tex] the angles of elevation meassured by each observer
S the area of the triangle formed with the observers and the ballonist
So, the area of a triangle is the length of its base times its height.
[tex]S=a*h[/tex] (equation 1)
but we can divide the triangle in two right triangles using the height line. So the total area will be equal to the addition of each individual area.
[tex]S=S_1+S_2[/tex] (equation 2)
[tex]S_1=a_1*h[/tex]
But we can write [tex]S_1[/tex] in terms of [tex]\alpha _1[/tex], like this:
[tex]\tan(\alpha _1)=\frac{h}{a_1} \\a_1=\frac{h}{\tan(\alpha _1)} \\S_1=\frac{h^{2} }{\tan(\alpha _1)}[/tex]
And for [tex]S_2[/tex] will be the same:
[tex]S_2=\frac{h^{2} }{\tan(\alpha _2)}[/tex]
Replacing in the equation 2:
[tex]S=\frac{h^{2} }{\tan(\alpha _1)}+\frac{h^{2} }{\tan(\alpha _2)}\\S=h^{2}*(\frac{1 }{\tan(\alpha _1)}+\frac{1}{\tan(\alpha _2)})[/tex]
And replacing in the equation 1:
[tex]h^{2}*(\frac{1 }{\tan(\alpha _1)}+\frac{1}{\tan(\alpha _2)})=a*h\\h=\frac{a}{(\frac{1 }{\tan(\alpha _1)}+\frac{1}{\tan(\alpha _2)})}[/tex]
So, we can replace all the known data in the last equation:
[tex]h=\frac{a}{(\frac{1 }{\tan(\alpha _1)}+\frac{1}{\tan(\alpha _2)})}\\h=\frac{7220 ft}{(\frac{1 }{\tan(35.6)}+\frac{1}{\tan(58.2)})}\\h=3579,91 ft[/tex]
Then, the ballonist is at a height of 3579.91 ft above the ground at 3:30pm.
