Question

to estimate the height of a building, two students find the angle of elevation from a point (at ground level) down the street from the building to the top of the building is 34°. from a point that is 400 feet closer to the building, the angle of elevation (at ground level) to the top of the building is 53°. if we assume that the street is level, use this information to estimate the height of the building.

the height of the building is \boxed{} feet.

question help: \boxed{video}

Explanation:

Step1: Set up equations using tangent function

Let the height of the building be \(h\) feet and the distance from the closer point to the building be \(x\) feet.
From the first - angle of elevation (\(\theta = 34^{\circ}\)), \(\tan34^{\circ}=\frac{h}{x + 400}\), so \(h=(x + 400)\tan34^{\circ}\).
From the second - angle of elevation (\(\theta = 53^{\circ}\)), \(\tan53^{\circ}=\frac{h}{x}\), so \(h = x\tan53^{\circ}\).

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

Since \((x + 400)\tan34^{\circ}=x\tan53^{\circ}\).
Expand: \(x\tan34^{\circ}+400\tan34^{\circ}=x\tan53^{\circ}\).
Rearrange: \(x(\tan53^{\circ}-\tan34^{\circ}) = 400\tan34^{\circ}\).
We know that \(\tan34^{\circ}\approx0.6745\) and \(\tan53^{\circ}\approx1.3270\).
Substitute the values: \(x(1.3270 - 0.6745)=400\times0.6745\).
\(x\times0.6525 = 269.8\).
Solve for \(x\): \(x=\frac{269.8}{0.6525}\approx413.5\).

Step3: Find the height \(h\)

Since \(h = x\tan53^{\circ}\), substitute \(x\approx413.5\) and \(\tan53^{\circ}\approx1.3270\).
\(h\approx413.5\times1.3270\approx550\).

Answer:

\(550\)