Question

from where kiana is standing, she must look up at a 50° angle to see the top of a building. if she backs up 50 feet, she will need to look up at a 40° angle to see the top of the same building. approximately how tall is the building?

Explanation:

Step1: Use trigonometric relations

Let the height of the building be \(h\) and the initial horizontal distance from Kiana to the building be \(x\).
We know that \(\tan50^{\circ}=\frac{h}{x}\), so \(h = x\tan50^{\circ}\).
Also, \(\tan40^{\circ}=\frac{h}{x + 50}\), so \(h=(x + 50)\tan40^{\circ}\).

Step2: Equate the two expressions for \(h\)

Since \(x\tan50^{\circ}=(x + 50)\tan40^{\circ}\).
We know that \(\tan50^{\circ}\approx1.19175\) and \(\tan40^{\circ}\approx0.8391\).
Substitute the values: \(1.19175x=0.8391(x + 50)\).
Expand: \(1.19175x=0.8391x+41.955\).
Subtract \(0.8391x\) from both sides: \(1.19175x-0.8391x = 41.955\).
\(0.35265x=41.955\).
Solve for \(x\): \(x=\frac{41.955}{0.35265}\approx119\).

Step3: Calculate the height \(h\)

Substitute \(x\approx119\) into \(h = x\tan50^{\circ}\).
\(h\approx119\times1.19175\approx142\) (feet).

Answer:

The height of the building is approximately \(142\) feet.