Question

1. a student wanted to know the distance between two particular carvings on a spirit pole. she measured the angle of elevation of each carving 15.0 m from the base of the pole. the student drew the sketch below. what is the distance between the carvings to the nearest tenth of a metre?

Explanation:

Step1: Find height of lower - carving

We use the tangent function in a right - triangle. For the lower carving with an angle of $30^{\circ}$ and adjacent side $x = 15.0$ m. The formula for tangent is $\tan\theta=\frac{opposite}{adjacent}$. So, $h_1=\tan(30^{\circ})\times15$. Since $\tan(30^{\circ})=\frac{\sqrt{3}}{3}$, then $h_1 = 15\times\frac{\sqrt{3}}{3}=5\sqrt{3}\approx8.66$ m.

Step2: Find height of upper - carving

For the upper carving with an angle of $45^{\circ}$ and adjacent side $x = 15.0$ m. Using $\tan\theta=\frac{opposite}{adjacent}$, and since $\tan(45^{\circ}) = 1$, then $h_2=\tan(45^{\circ})\times15=15$ m.

Step3: Calculate distance between carvings

The distance $d$ between the carvings is $d=h_2 - h_1$. Substitute the values of $h_1$ and $h_2$: $d = 15-5\sqrt{3}\approx15 - 8.66 = 6.34\approx6.3$ m.

Answer:

$6.3$ m