Question

A flagpole six meters tall stands on a sloping roof. A wire seven meters long joins the top of the flagpole to the roof at a point 8 meters up from the base of the pole. What angle does the roof make with the horizontal?

Answer 1

By applying trigonometric principles and using the cosine function, we can calculate the angle formed by the roof with the horizontal relative to the flagpole.

Step-by-step explanation:

To calculate the angle the roof makes with the horizontal, we can use trigonometry. We're given a flagpole that is 6 meters tall, and a wire that is 7 meters long, which joins the top of the flagpole to the roof at a point 8 meters up from the base of the pole. We're dealing with a right-angled triangle, where the flagpole is the vertical side, the wire is the hypotenuse, and the distance up the roof is the base of the triangle. To find the angle the roof makes with the horizontal, we need to find the angle between the wire and the vertical flagpole, which can be done using the cosine formula:

\(\cos(\theta) = \frac{adjacent}{hypotenuse}\)

\(\cos(\theta) = \frac{6}{7}\)

Now solving for \(\theta\) we get that \(\theta\) is approximately equal to \(\cos^{-1}(\frac{6}{7})\) degrees. Therefore, the angle the roof makes with the horizontal is \(90^\circ - \theta\).