Question

A student wants to know how far above the ground the top of a leaning flagpole is. At high noon, when the sun is almost directly overhead, the shadow cast by the pole is 5 ft long. The student holds a plumb bob with a string 3 ft long up to the flagpole and determines that the point of the plumb bob touches the ground 11 in. from the base of the flagpole. How far above the ground is the top of the pole?

Answer 1

Answer:

The top of the pole is about  16.36 ft above the ground.

Step-by-step explanation:

The problem can be addressed by using similar triangles. In this case we have two right angle triangles for which we have three known values and the requested unknown (please see accompanying figure)

The larger triangle is formed with the actual flag-pole, the 5 ft cast shadow (represented in green in the image), and the height from the ground the top of the flag-pole is (represented in red and with a letter "x" in the image).

The second smaller triangle is that formed with the 3ft long plumb bob that touches the ground at 11 inches from the base of the flag-pole (this smaller triangle is represented in blue in the attached image).

Both triangles share the same angle that the flag-pole makes with the ground, and are similar triangles. Therefore the ratio between their correspondent sides must be a constant. That is: The base "B" of the larger triangle is to the base "b" of the smaller one, in the same ratio as the height "H" of the larger triangle is to the heigh "h" of the smaller one:

$\frac{B}{b}=\frac{H}{h}$

Before creating this ratio equation with our values, we take care of reducing all known sizes to the SAME units, so comparing the ratios makes sense.

We select to write all magnitudes using inches, therefore:

B = 5 * 12 = 60 in

b = 11 in

h = 3 * 12 = 36 in

and then, the unknown height of the larger triangle (which we named "x") will result also in inches.

We now set the ratio equation using these magnitudes:

$\frac{B}{b}=\frac{H}{h}\\\frac{60}{11}=\frac{x}{36}\\x=\frac{60*36}{11} \\x=\frac{2160}{11} \\x=196.36\,in$

We can express this quantity in feet by dividing it by 12:

$x=16.36\, ft$