The tops of two poles of height 40 m and 25 m are connected by a wire of length `(30sqrt2)/((sqrt3-1))m`. If the wire makes an angle `alpha` with the horizontal, then the value of `sqrt2sin alpha` is equal to `("take, "sqrt3=1.7)`

Answer

To solve the problem step by step, we will first visualize the situation and then apply trigonometric principles to find the required value.

### Step 1: Visualize the Problem
We have two poles of heights 40 m and 25 m. Let's denote:
- The height of the first pole (A) = 40 m
- The height of the second pole (B) = 25 m

The vertical distance between the tops of the two poles is:
\[ \text{Height difference} = 40 \, \text{m} - 25 \, \text{m} = 15 \, \text{m} \]

### Step 2: Identify the Length of the Wire
The length of the wire connecting the tops of the two poles is given as:
\[ L = \frac{30\sqrt{2}}{\sqrt{3} - 1} \, \text{m} \]

### Step 3: Set Up the Right Triangle
Let:
- Point O be the top of the first pole (height 40 m)
- Point P be the top of the second pole (height 25 m)
- Point Q be the point on the ground directly below the wire connection

In triangle OPQ:
- OP (the vertical height difference) = 15 m
- PQ (the length of the wire) = L

### Step 4: Apply Trigonometric Ratios
Using the sine function, we have:
\[ \sin(\alpha) = \frac{OP}{PQ} = \frac{15}{L} \]

Substituting the value of L:
\[ \sin(\alpha) = \frac{15}{\frac{30\sqrt{2}}{\sqrt{3} - 1}} \]

### Step 5: Simplify the Expression
To simplify:
\[ \sin(\alpha) = \frac{15(\sqrt{3} - 1)}{30\sqrt{2}} \]
\[ \sin(\alpha) = \frac{\sqrt{3} - 1}{2\sqrt{2}} \]

### Step 6: Find the Value of \( \sqrt{2} \sin(\alpha) \)
Now, we need to find \( \sqrt{2} \sin(\alpha) \):
\[ \sqrt{2} \sin(\alpha) = \sqrt{2} \cdot \frac{\sqrt{3} - 1}{2\sqrt{2}} \]
\[ \sqrt{2} \sin(\alpha) = \frac{\sqrt{3} - 1}{2} \]

### Step 7: Substitute the Value of \( \sqrt{3} \)
Given \( \sqrt{3} = 1.7 \):
\[ \sqrt{2} \sin(\alpha) = \frac{1.7 - 1}{2} \]
\[ \sqrt{2} \sin(\alpha) = \frac{0.7}{2} \]
\[ \sqrt{2} \sin(\alpha) = 0.35 \]

### Final Answer
Thus, the value of \( \sqrt{2} \sin(\alpha) \) is \( 0.35 \).

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