Question

from her eye, which stands 1.75 meters above the ground, myranda measures the angle of elevation to the top of a prominent skyscraper to be 19°. if she is standing at a horizontal distance of 197 meters from the base of the skyscraper, what is the height of the skyscraper? round your answer to the nearest hundredth of a meter if necessary.

Explanation:

Step1: Identify the trigonometric relationship

We can use the tangent function in a right triangle. The angle of elevation is \(59^\circ\), the horizontal distance (adjacent side) is \(397\) meters, and we need to find the vertical height from Myranda's eye level to the top of the skyscraper (opposite side), let's call this height \(h\). The formula for tangent is \(\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}\), so \(\tan(59^\circ)=\frac{h}{397}\).

Step2: Solve for \(h\)

Multiply both sides by \(397\) to get \(h = 397\times\tan(59^\circ)\). Using a calculator, \(\tan(59^\circ)\approx1.6643\), so \(h\approx397\times1.6643\approx660.73\) meters.

Step3: Add Myranda's eye height

Myranda's eye is \(1.79\) meters above the ground, so the total height of the skyscraper \(H=h + 1.79\approx660.73+1.79 = 662.52\) meters.

Answer:

\(662.52\)