Sure! Let's solve the problem step-by-step using trigonometry.
### Given:
- Height of the tower ([tex]\( h_{\text{tower}} \)[/tex]) = 80 meters
- Angle of elevation from the top of the building to the top of the tower ([tex]\( \theta_{\text{top}} \)[/tex]) = 35 degrees
- Angle of elevation from the bottom of the building to the top of the tower ([tex]\( \theta_{\text{bottom}} \)[/tex]) = 65 degrees
### To Find:
(a) The distance of the building from the tower (denoted as [tex]\( d \)[/tex]).
(b) The height of the building (denoted as [tex]\( h_{\text{building}} \)[/tex]).
### Step-by-Step Solution:
#### Part (a): Calculating the Distance [tex]\( d \)[/tex]
1. Use the angle of elevation from the bottom of the building:
The angle of elevation from the bottom of the building to the top of the tower forms a right triangle with the distance between the building and the tower being the adjacent side and the height of the tower as the opposite side.
We can use the tangent function:
[tex]\[
\tan(65^\circ) = \frac{h_{\text{tower}}}{d}
\][/tex]
2. Solve for [tex]\( d \)[/tex]:
[tex]\[
d = \frac{h_{\text{tower}}}{\tan(65^\circ)}
\][/tex]
Substitute the given values:
[tex]\[
d = \frac{80}{\tan(65^\circ)} \approx 37.30 \text{ meters}
\][/tex]
So, the distance of the building from the tower is approximately [tex]\( 37.30 \)[/tex] meters.
#### Part (b): Calculating the Height [tex]\( h_{\text{building}} \)[/tex]
1. Use the angle of elevation from the top of the building:
The height difference between the top of the tower and the top of the building is [tex]\( h_{\text{tower}} - h_{\text{building}} \)[/tex]. This forms another right triangle with the same distance [tex]\( d \)[/tex] as the adjacent side and this height difference as the opposite side.
We can use the tangent function again:
[tex]\[
\tan(35^\circ) = \frac{h_{\text{tower}} - h_{\text{building}}}{d}
\][/tex]
2. Solve for [tex]\( h_{\text{building}} \)[/tex]:
[tex]\[
h_{\text{tower}} - h_{\text{building}} = d \cdot \tan(35^\circ)
\][/tex]
Substitute the value of [tex]\( d \)[/tex]:
[tex]\[
h_{\text{tower}} - h_{\text{building}} = 37.30 \cdot \tan(35^\circ)
\][/tex]
Calculate:
[tex]\[
h_{\text{tower}} - h_{\text{building}} \approx 26.12
\][/tex]
Now, solve for [tex]\( h_{\text{building}} \)[/tex]:
[tex]\[
h_{\text{building}} = h_{\text{tower}} - 26.12
\][/tex]
Substitute [tex]\( h_{\text{tower}} \)[/tex]:
[tex]\[
h_{\text{building}} = 80 - 26.12 \approx 53.88 \text{ meters}
\][/tex]
So, the height of the building is approximately [tex]\( 53.88 \)[/tex] meters.
