Question

from a window 31.0 ft above the street, the angle of elevation to the top of the building across the street is 46.0° and the angle of depression to the base of this building is 17.0°. find the height of the building across the street.
the height of the building across the street is \\( \square \\) ft.
(round to the nearest whole number as needed.)

Explanation:

Step1: Find the horizontal distance (x)

The angle of depression to the base is \(17.0^\circ\), and the vertical distance from the window to the street is \(31.0\) ft. Using the tangent function for the angle of depression: \(\tan(17.0^\circ)=\frac{31.0}{x}\), so \(x = \frac{31.0}{\tan(17.0^\circ)}\).
\(\tan(17.0^\circ)\approx0.3057\), so \(x=\frac{31.0}{0.3057}\approx101.4\) ft.

Step2: Find the vertical distance (y) from the window to the top of the building

Using the angle of elevation \(46.0^\circ\) and the horizontal distance \(x\), \(\tan(46.0^\circ)=\frac{y}{x}\), so \(y = x\tan(46.0^\circ)\).
\(\tan(46.0^\circ)\approx1.0355\), so \(y = 101.4\times1.0355\approx105.0\) ft.

Step3: Find the total height of the building

The total height \(H\) is the sum of the window height and \(y\): \(H = 31.0 + 105.0 = 136.0\) ft (wait, recalculating with more precise \(x\)):

First, more precise \(x\): \(\tan(17^\circ)=\tan(17\times\frac{\pi}{180})\approx0.305730681\), so \(x=\frac{31}{0.305730681}\approx101.396\) ft.

Then \(y = 101.396\times\tan(46^\circ)\), \(\tan(46^\circ)\approx1.035530314\), so \(y\approx101.396\times1.035530314\approx105.0\) (more precisely, \(101.396\times1.035530314 = 101.396\times1 + 101.396\times0.035530314\approx101.396 + 3.602\approx104.998\approx105\) ft).

Total height \(H = 31 + 105 = 136\) ft? Wait, no, wait: Wait, the angle of depression: the horizontal line from the window to the building, the vertical distance from window to street is 31 ft, so the triangle for angle of depression has opposite side 31, adjacent side x, so \(\tan(17^\circ)=\frac{31}{x}\), so \(x = 31 / \tan(17^\circ)\). Then the angle of elevation: the triangle from window to top of building has opposite side y (height above window) and adjacent side x, so \(\tan(46^\circ)=\frac{y}{x}\), so \(y = x \tan(46^\circ)\). Then total height is \(31 + y\).

Let's calculate with more precision:

\(\tan(17^\circ) \approx 0.30573068145\)

\(x = 31 / 0.30573068145 \approx 101.396\) ft

\(\tan(46^\circ) \approx 1.03553031379\)

\(y = 101.396 \times 1.03553031379 \approx 101.396 \times 1.03553 \approx 101.396 + 101.396\times0.03553 \approx 101.396 + 3.602 \approx 104.998 \approx 105\) ft

Total height: \(31 + 105 = 136\) ft? Wait, but let's check again. Wait, maybe I made a mistake. Wait, the angle of depression is 17 degrees, so the angle between the horizontal and the line of sight to the base is 17 degrees, so the triangle is right-angled, with opposite side 31 (window height), adjacent side x (distance between buildings). So \(\tan(17^\circ) = 31 / x\), so \(x = 31 / \tan(17^\circ)\). Then the angle of elevation is 46 degrees, so the height above the window is \(y = x \tan(46^\circ)\). Then total height is \(31 + y\).

Calculating \(x\): \(31 / \tan(17^\circ) = 31 / 0.30573068145 \approx 101.396\)

\(y = 101.396 \times \tan(46^\circ) = 101.396 \times 1.03553031379 \approx 105.0\)

So total height: \(31 + 105 = 136\) ft? Wait, but let's use a calculator for more precision.

Wait, \(\tan(17^\circ) \approx 0.30573068145\), so \(x = 31 / 0.30573068145 \approx 101.396\)

\(\tan(46^\circ) \approx 1.03553031379\), so \(y = 101.396 * 1.03553031379 = 101.396 * 1.03553031379\)

Let's compute 101.396 * 1.03553031379:

101.396 * 1 = 101.396

101.396 * 0.03 = 3.04188

101.396 * 0.00553031379 ≈ 101.396 * 0.005 = 0.50698, 101.396 * 0.00053031379 ≈ 0.05377, so total ≈ 0.50698 + 0.05377 ≈ 0.56075

So total y ≈ 101.396 + 3.04188 + 0.56075 ≈ 104.99863 ≈ 105.0

Then total height: 31 + 105 = 136 ft. Wait, but maybe I messed up the angle of depression. Wait, angle of depression is equal to the angle of elevation from the base to the window, so that part is correct. So the horizontal distance is x, window height is 31, so the building across has height 31 + y, where y is the height from window to top. So yes, that makes sense.

Wait, but let's check with another approach. Let's use the two right triangles. The first triangle (depression) has angle 17°, opposite 31, adjacent x. The second triangle (elevation) has angle 46°, opposite y, adjacent x. So x is common. So x = 31 / tan(17°), y = x tan(46°) = 31 tan(46°) / tan(17°). Then total height H = 31 + y = 31(1 + tan(46°)/tan(17°)).

Calculating tan(46°)/tan(17°): 1.03553031379 / 0.30573068145 ≈ 3.387

Then 31 * 3.387 ≈ 104.997, then H = 31 + 104.997 ≈ 135.997 ≈ 136 ft.

Yes, so the total height is approximately 136 ft.

Answer:

\(136\)