From a point on the floor the angle of elevation to the top of a door is 47°, while the angle of elevation to the ceiling above the door is 62°. If the ceiling is 10 feet above the floor, what is the vertical dimension of the door (see the figure). Round your answer to the nearest tenth of a feet.

Question

From a point on the floor the angle of elevation to the top of a door is 47°, while the angle of elevation to the ceiling above the door is 62°. If the ceiling is 10 feet above the floor, what is the vertical dimension of the door (see the figure). Round your answer to the nearest tenth of a feet.   ​

Quizplus · Accepted Answer

Let the height of the door be $x$ feet. Then, we have the following diagram:

[asy]
pair A,B,C,P;
A=(0,0);
B=(2.5,0);
P=(2.5,tan(62));
C=(0,tan(47));
draw(A--P--B--cycle);
draw(P--C);
label("$10$",(A+C)/2,W);
label("$x$",(C+P)/2,NE);
label("$h-x$",(P+B)/2,NE);
label("$47^\circ$",A,SE);
label("$62^\circ$",B,SW);
[/asy]

From the diagram, we can see that $	an 47^\circ = \frac{10}{x}$ and $	an 62^\circ = \frac{10}{h-x}$. Solving for $x$ in the first equation, we get $x = \frac{10}{	an 47^\circ}$. Substituting this expression for $x$ in the second equation, we get $	an 62^\circ = \frac{10}{h-\frac{10}{	an 47^\circ}}$. Solving for $h$, we get $h = \frac{10}{	an 62^\circ} + \frac{10}{	an 47^\circ} \approx 15.692 \approx \boxed{5.7	ext{ ft}}$ (rounded to the nearest tenth).

From a Point on the Floor the Angle of Elevation