Certainly! Let's solve the problem step-by-step:
Given:
1. The angle of elevation to the top of the CN Tower from the first point is 65°.
2. From a position 56.6 meters closer, the angle of elevation is 70°.
Let's denote:
- [tex]\( h \)[/tex] as the height of the CN Tower.
- [tex]\( d \)[/tex] as the horizontal distance from the original point to the base of the tower.
### Step 1: Establish the trigonometric relationships
From the first point:
Using the angle of elevation of 65°, we can write:
[tex]\[ \tan(65^\circ) = \frac{h}{d} \][/tex]
Thus,
[tex]\[ h = d \cdot \tan(65^\circ) \][/tex]
From the closer point:
Now, 56.6 meters closer to the tower, the angle of elevation is 70°. Therefore:
[tex]\[ \tan(70^\circ) = \frac{h}{d - 56.6} \][/tex]
Thus,
[tex]\[ h = (d - 56.6) \cdot \tan(70^\circ) \][/tex]
### Step 2: Solve for [tex]\( d \)[/tex]
We have two expressions for [tex]\( h \)[/tex]:
[tex]\[ d \cdot \tan(65^\circ) = (d - 56.6) \cdot \tan(70^\circ) \][/tex]
Let's set up the equation:
[tex]\[ d \cdot \tan(65^\circ) = d \cdot \tan(70^\circ) - 56.6 \cdot \tan(70^\circ) \][/tex]
Rearrange to isolate [tex]\( d \)[/tex]:
[tex]\[ d \cdot \tan(65^\circ) - d \cdot \tan(70^\circ) = -56.6 \cdot \tan(70^\circ) \][/tex]
Factor [tex]\( d \)[/tex] from the left side:
[tex]\[ d \cdot (\tan(65^\circ) - \tan(70^\circ)) = -56.6 \cdot \tan(70^\circ) \][/tex]
Finally, solve for [tex]\( d \)[/tex]:
[tex]\[ d = \frac{-56.6 \cdot \tan(70^\circ)}{\tan(65^\circ) - \tan(70^\circ)} \][/tex]
### Step 3: Compute the value of [tex]\( d \)[/tex]
Given the calculations,
[tex]\[ d \approx 257.9 \, \text{meters} \][/tex]
### Step 4: Compute the height [tex]\( h \)[/tex]
Now, using the initial tangent relation to find [tex]\( h \)[/tex]:
[tex]\[ h = d \cdot \tan(65^\circ) \][/tex]
[tex]\[ h \approx 257.9 \cdot \tan(65^\circ) \][/tex]
Performing this calculation,
[tex]\[ h \approx 553.1 \, \text{meters} \][/tex]
### Conclusion
The height of the CN Tower is approximately 553.1 meters to the nearest tenth of a meter.
