The height of the tower will be given by:
[tex]\tan\theta=\frac{opposite}{adjacent}[/tex]
Where:
opposite = h
adjacent = x
tetha = 67°
Therefore:
[tex]\tan67=\frac{h}{x}[/tex]
Solve for h:
[tex]\begin{gathered} x\cdot\tan67=x\cdot\frac{h}{x} \\ h=x\cdot\tan67 \end{gathered}[/tex]
And for the other angle:
[tex]\tan39=\frac{h}{x+32}[/tex]
Solve for h:
[tex]h=(x+32)\cdot\tan39[/tex]
Equating the two equations for h, we have:
[tex]x\tan67=(x+32)\tan39[/tex]
Now, solve for x:
[tex]\begin{gathered} x(2.36)=(x+32)(0.81) \\ 2.36x=0.810x+25.92 \\ 2.36x-0.81x=0.81x+25.92-0.81x \\ 1.55x=25.92 \\ \frac{1.55x}{1.55}=\frac{25.92}{1.55} \\ x=16.72\approx17 \end{gathered}[/tex]
Now, we find h:
[tex]h=17\cdot\tan67=40.05\approx40[/tex]
Answer: A 40 ft
