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 °.

(b) How high is the window?

Answer 1

To find the height of the window (h), we can use the information given about the rope and the angle it makes with the side of the building. The rope essentially forms a right triangle with the wall of the building and the ground.

In this right triangle:

- The height of the window (h) is the vertical side opposite the angle.
- The distance from the base of the building to the point on the ground is the horizontal side, which is 6 m.
- The angle between the rope and the side of the building is 19°.

Using the tangent function, which is defined as the opposite side divided by the adjacent side, we have:

\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]

Substituting in the values we have:

\[
\tan(19^\circ) = \frac{h}{6}
\]

Now we can solve for \( h \):

\[
h = 6 \times \tan(19^\circ)
\]

Now, we need to calculate \( \tan(19^\circ) \):

\(\tan(19^\circ) \approx 0.3443\) (using a calculator or trigonometric table).

Substituting this value back into the equation for \( h \):

\[
h \approx 6 \times 0.3443 \approx 2.0658 \text{ m}
\]

So, rounding to two decimal places, the height of the window is approximately:

\[
\boxed{2.07 \text{ m}}
\]