The height of the tower is 88 ft
Here, we want to get the height of the tower
We proceed as follows;
Let us represent this height by h
We can also have the full distance of the base as (100 + x)
What this mean is that for the triangle that contains the height and the angle 65, the base is x
Now, we can use the appropriate trigonometric identities to link up
With the angle 65, the tower is the opposite, while x is the adjacent
We use the tangent here;
[tex]\begin{gathered} \tan \text{ 65 = }\frac{h}{x} \\ h\text{ = xtan65} \end{gathered}[/tex]
Furthermore, we have it that the base (100+x) is the adjacent and the height h is the oppsoite for the triangle that contains the angle 32
Thus, we have it that;
[tex]\begin{gathered} \tan \text{ 32 = }\frac{h}{100\text{ + x}} \\ \\ h\text{ = tan 32(100+x)} \end{gathered}[/tex]
From here, we equate the two h values;
[tex]\begin{gathered} x\text{ tan 65 = tan 32(100+x)} \\ 2.1445x\text{ = 0.62487(100+x)} \\ 2.1445x\text{ = 62.487+0.62487x} \\ 2.1445x-0.62487x\text{ = 62.487} \\ 1.51963x\text{ = 62.487} \\ x\text{ = }\frac{62.487}{1.51963} \\ \\ x\text{ = 41.12 ft} \end{gathered}[/tex]
To get the value of h, we simply substitute;
[tex]\begin{gathered} h\text{ = x tan 65} \\ h\text{ = 41.12 }\times2.1445 \\ h\text{ = 88.18 ft} \end{gathered}[/tex]
