Absolutely, let's work through this mathematical problem step-by-step!
We are given:
- The distance between two buildings: 10 meters,
- The angle of depression from the top of the shorter building (Y) to the foot of the taller building (P): 53°,
- The angle of elevation from the top of the shorter building to the top of the taller building (X): 28°.
Let's first summarize the situation:
1. Let [tex]\( h_Y \)[/tex] be the height of the shorter building.
2. Let [tex]\( h_{\text{tall}} \)[/tex] be the height of the taller building.
3. Let [tex]\( h_d \)[/tex] be the height difference between the top of the taller building and the top of the shorter building.
### Step 1: Determine the height of the shorter building (h_Y)
To determine [tex]\( h_Y \)[/tex], we'll use the tangent of the angle of depression:
[tex]\[
\tan(53^\circ) = \frac{h_Y}{\text{distance between buildings}}
\][/tex]
Given that the distance between the buildings is 10 meters, we have:
[tex]\[
\tan(53^\circ) = \frac{h_Y}{10}
\][/tex]
Solving for [tex]\( h_Y \)[/tex], we get:
[tex]\[
h_Y = 10 \times \tan(53^\circ)
\][/tex]
Using a calculator to find [tex]\( \tan(53^\circ) \)[/tex]:
[tex]\[
h_Y = 10 \times 1.327
\][/tex]
[tex]\[
h_Y \approx 13.27 \text{ meters}
\][/tex]
### Step 2: Determine the height difference (h_d)
Next, we use the angle of elevation to find the height difference [tex]\( h_d \)[/tex] between the tops of the two buildings:
[tex]\[
\tan(28^\circ) = \frac{h_d}{\text{distance between buildings}}
\][/tex]
Given the distance between the buildings is still 10 meters, we have:
[tex]\[
\tan(28^\circ) = \frac{h_d}{10}
\][/tex]
Solving for [tex]\( h_d \)[/tex], we get:
[tex]\[
h_d = 10 \times \tan(28^\circ)
\][/tex]
Using a calculator to find [tex]\( \tan(28^\circ) \)[/tex]:
[tex]\[
h_d = 10 \times 0.5317
\][/tex]
[tex]\[
h_d \approx 5.317 \text{ meters}
\][/tex]
### Step 3: Calculate the height of the taller building (h_{\text{tall}})
Finally, to find the height of the taller building, add the height of the shorter building [tex]\( h_Y \)[/tex] to the height difference [tex]\( h_d \)[/tex]:
[tex]\[
h_{\text{tall}} = h_Y + h_d
\][/tex]
[tex]\[
h_{\text{tall}} = 13.27 + 5.317
\][/tex]
[tex]\[
h_{\text{tall}} \approx 18.587 \text{ meters}
\][/tex]
Thus, the height of the taller building is approximately 18.587 meters.
