Question

the vertical angle from level ground to the top of a building is 40°. the angle is measured from a point that is 25 m distant from the base of the building. how tall is the building?

Explanation:

Step1: Identify the trigonometric relationship

We have a right triangle where the adjacent side to the \(40^\circ\) angle is \(25\) m (distance from the point to the base of the building), and the opposite side is the height \(h\) of the building. We use the tangent function, which is defined as \(\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}\).
So, \(\tan(40^\circ)=\frac{h}{25}\)

Step2: Solve for \(h\)

To find \(h\), we multiply both sides of the equation by \(25\):
\(h = 25\times\tan(40^\circ)\)
We know that \(\tan(40^\circ)\approx0.8391\) (using a calculator).
So, \(h = 25\times0.8391 = 20.9775\)

Answer:

The building is approximately \(21.0\) m tall (or \(20.98\) m depending on the level of precision).