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? 119.0 ft 141.8 ft 185.1 ft 220.6 ft

Explanation:

Step1: Set up equations using tangent function

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 \(\tan40^{\circ}\approx0.839099\).
So \(1.19175x=0.839099(x + 50)\).
Expand the right - hand side: \(1.19175x=0.839099x+41.95495\).
Subtract \(0.839099x\) from both sides: \(1.19175x-0.839099x=41.95495\).
\(0.352651x = 41.95495\).
Solve for \(x\): \(x=\frac{41.95495}{0.352651}\approx119\).

Step3: Calculate the height \(h\)

Substitute \(x\approx119\) into \(h = x\tan50^{\circ}\).
\(h\approx119\times1.19175\approx141.8\)

Answer:

\(141.8\space ft\)