Question

a. draw a detailed picture of this situation. label it with all of the given information.
step 1(a):
draw leon, the telephone pole, and the hawk.
step 2(a):
draw sight and distance lines.
step 3(a):
label the known distance(s) and angle(s). label the unknown
height of the telephone pole ( x ).
b. how tall is the pole? show all of your work.
hint (b):
use the tangent ratio to write an equation and solve for ( x ).
( \tan(22^{circ}) = (\frac{x}{60}) )
answer (b):
( 60 cdot \tan(22^{circ}) = x )
( 24.24 approx x )
height of pole ( approx 24.24 + 5.2 approx 29.44 ) feet

Explanation:

Step1: Use the tangent function

In a right - triangle (formed by the line of sight, the horizontal line, and the vertical part of the pole above Leon's height), the tangent of an angle of elevation is given by the formula \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). Here, \(\theta = 22^{\circ}\), the adjacent side to the angle \(22^{\circ}\) is \(60\) (horizontal distance), and the opposite side is \(x\) (the part of the pole above Leon's height). So, \(\tan(22^{\circ})=\frac{x}{60}\).

Step2: Solve for \(x\)

Multiply both sides of the equation \(\tan(22^{\circ})=\frac{x}{60}\) by \(60\) to isolate \(x\). We know that \(\tan(22^{\circ})\approx0.404\). Then \(x = 60\times\tan(22^{\circ})\approx60\times0.404 = 24.24\).

Step3: Find the total height of the pole

The total height of the pole \(h\) is the sum of the part above Leon's height (\(x\)) and Leon's height (\(5.2\)). So \(h=x + 5.2\). Substitute \(x\approx24.24\) into the formula, we get \(h\approx24.24+5.2\).

Answer:

The height of the pole is approximately \(29.44\) feet.