Sure, let's solve this problem step by step.
### Step 1: Understanding the problem
- We have a radio mast that is 15 cm high.
- There are two huts, one to the east of the mast and one to the west.
- The angle of depression from the top of the mast to the east hut is [tex]\(60^\circ\)[/tex].
- The angle of depression from the top of the mast to the west hut is [tex]\(45^\circ\)[/tex].
- We need to find the distance between the two huts.
### Step 2: Trigonometric approach
We will use trigonometry to find the distances from the mast to each of the huts. The mast height (opposite side of the angle) is 15 cm, and we use the tangent function, which relates an angle to the opposite and adjacent sides in a right triangle.
### Step 3: Calculate distances using tangent function
1. East Hut Distance Calculation:
[tex]\[
\tan(60^\circ) = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
Here, the opposite side is the mast height (15 cm), and the adjacent side is the distance to the east hut ([tex]\(d_{\text{east}}\)[/tex]).
[tex]\[
\tan(60^\circ) = \frac{15}{d_{\text{east}}}
\][/tex]
Solving for [tex]\(d_{\text{east}}\)[/tex]:
[tex]\[
d_{\text{east}} = \frac{15}{\tan(60^\circ)}
\][/tex]
2. West Hut Distance Calculation:
[tex]\[
\tan(45^\circ) = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
Here, the opposite side is again the mast height (15 cm), and the adjacent side is the distance to the west hut ([tex]\(d_{\text{west}}\)[/tex]).
[tex]\[
\tan(45^\circ) = \frac{15}{d_{\text{west}}}
\][/tex]
Solving for [tex]\(d_{\text{west}}\)[/tex]:
[tex]\[
d_{\text{west}} = \frac{15}{\tan(45^\circ)}
\][/tex]
### Step 4: Substitute the values of tangent
- [tex]\(\tan(60^\circ) = \sqrt{3} \approx 1.732\)[/tex]
- [tex]\(\tan(45^\circ) = 1\)[/tex]
So:
[tex]\[
d_{\text{east}} = \frac{15}{1.732} \approx 8.66 \text{ cm}
\][/tex]
[tex]\[
d_{\text{west}} = \frac{15}{1} = 15 \text{ cm}
\][/tex]
### Step 5: Calculate the total distance between the huts
The total distance between the two huts is the sum of the distances from the mast to each hut:
[tex]\[
\text{Total Distance} = d_{\text{east}} + d_{\text{west}}
\][/tex]
Substituting the values we found:
[tex]\[
\text{Total Distance} = 8.66 \text{ cm} + 15 \text{ cm} = 23.66 \text{ cm}
\][/tex]
### Conclusion
Thus, the distance between the two huts is approximately 23.66 cm. The distances from the mast to the east and west huts are approximately 8.66 cm and 15 cm, respectively.
