To answer this question, it would make it easier if we graph the information as follows:
Now, we can observe that, in conjunction with the angles of depression and elevation, we can find the value for the total height of the Radio Tower.
We can use trigonometric ratios to find them. We can also see that the shared side of the triangle (the green line) is equal to 600 feet.
Then, to find h1, we can proceed as follows:
[tex]\tan (\theta)=\frac{opp}{adj}\Rightarrow\tan (32)=\frac{h_1}{600}\Rightarrow h_1=600\cdot\tan (32)[/tex]And
[tex]\tan (25)=\frac{h_2}{600}\Rightarrow h_2=600\cdot\tan (25)[/tex]Therefore, the total height is given by the sum of both heights (h1 + h2). Then, we have:
[tex]h_1=600\cdot\text{tan}(32)=600\cdot0.624869351909=$374.921611146$ft[/tex]And
[tex]h_2=600\cdot\tan (25)=600\cdot0.466307658155=279.784594893ft[/tex]Then, the total height of the Radio Tower is:
[tex]h_{\text{RadioTower}}=374.921611145ft+279.784594893ft=654.706206038ft[/tex]If we round our answer to four decimal places, we have that the height of the Radio Tower is:
[tex]h_{\text{RadioTower}}=654.7062ft[/tex]In summary, the height of the Radio Tower is 654.7062 feet.
