Question

Pat needs to determine the height of a tree before cutting it down to be sure that it will not fall on a nearby fence. the angle of elevation of the tree from one position on a flat path from the tree is upper h equals 30 degrees commah=30°, and from a second position upper l equals 50 feetl=50 feet farther along this path it is upper b equals 20 degrees .b=20°. what is the height of the​ tree?

Answer 1

Please find the attached diagram for a better understanding of the solution provided here.

In the attached diagram, OP is the height of the tree.

Rest of the diagram should be sufficiently self explanatory.

In $\Delta OPH$,

$tan(30^0)=\frac{OP}{x}$

$\therefore \frac{OP}{tan(30^0)}=x$.........................Equation #1

Again, in $\Delta OPL$,

$tan(20^0)=\frac{OP}{PH+HL}=\frac{OP}{x+50}$

$\therefore \frac{OP}{tan(20^0)}-50=x$ .........................Equation #2

Thus, equating the equations Equation #1 and Equation #2, we have:

$\frac{OP}{tan(30^0)}=\frac{OP}{tan(20^0)}-50$

Evaluating we get:

$1.732(OP)=2.747(OP)-50$

$1.015(OP)=50$

$\therefore OP=\frac{50}{1.015}\approx49.26 ft$

Thus the height of the tree is approximately 49.26 feet