Chapter 6, Problem 44CLT

To determine

To identify: The height of the flag to the nearest foot.

Answer to Problem 44CLT

The height of flag is 5 ft.

Explanation of Solution

Given Information:

From a point 200 feet from a flagpole, the angles of elevation to the bottom and top of the flag are ${16}^{\circ }45^{\prime }$ and ${18}^{\circ }$ degree.

Concept Involved:

Use the trigonometric formula such that

  $\tan \theta =\frac{\text{Perpendicular}}{\text{Base}}$

Consider h be the height of flag.

  

In $\Delta BDC,$

  $\begin{array}{l}\tan {16}^{\circ }45^{\prime }=\frac{BC}{CD}\\ BC=CD\tan {16}^{\circ }45^{\prime }\\ BC=200\tan {16}^{\circ }45^{\prime }\end{array}$

In $\Delta ADC,$

  $\begin{array}{l}\tan {16}^{\circ }45^{\prime }=\frac{AC}{CD}\\ AC=CD\tan {18}^{\circ }\\ h+BC=200\tan {18}^{\circ }\end{array}$

Substitute the value of $BC$ ,

  $\begin{array}{l}h+200\tan {16}^{\circ }45^{\prime }=200\tan {18}^{\circ }\\ h=200\tan {18}^{\circ }-200\tan {16}^{\circ }45^{\prime }\\ h=200\left[\tan {18}^{\circ }-\tan {16}^{\circ }45^{\prime }\right]\\ h=200\left[0.32491969-0.30096584\right]\\ h=200\times 0.02395385\\ h=4.79077\approx 5\text{ft}\end{array}$

Conclusion:

The height of flag is 5 ft.