Question

A rope is stretched from a window in the side of a building to a point on the ground, 6 m from the base of the building. The angle between the rope and the side of the building is 19 °.


(a) How long is the rope?

Answer 1

To find the length of the rope, we can use trigonometric relationships. In this case, we can use the sine, cosine, or tangent functions to relate the angle, the height of the building (which we need to find), and the distance from the base of the building to the point on the ground.

Let's denote:

- \( L \) = length of the rope (hypotenuse)
- \( h \) = height of the building (opposite side of the triangle)
- \( d \) = distance from the base of the building to the point on the ground = 6 m
- \( \theta \) = angle between the rope and the side of the building = 19°

Using the cosine function, we have:

\[
\cos(\theta) = \frac{d}{L}
\]

From above:

\[
L = \frac{d}{\cos(\theta)}
\]

Now plug in the values:

\[
L = \frac{6}{\cos(19°)}
\]

To calculate \( \cos(19°) \), we can use a calculator:

\[
\cos(19°) \approx 0.9455
\]

Substituting this value back into the formula gives:

\[
L \approx \frac{6}{0.9455} \approx 6.34 \, \text{m}
\]

Thus, the length of the rope is approximately \( 6.34 \) meters.