A circus artist is climbing from the ground along a rope stretched from the top of a vertical pole and tied at the ground. The height of the pole is 10 m and angle made by the top with ground level is `60^(@)` .
(i) Calculate the distance covered by the artist in climbing to the top of the pole.
(ii) Which mathematical concept is used in this problem ?
(iii) What is its value ?

Answer

To solve the problem step by step, we will use trigonometric concepts.

### Step 1: Understand the Problem
We have a vertical pole of height 10 m and a rope tied from the top of the pole to the ground, making an angle of 60 degrees with the ground. We need to find the distance covered by the artist in climbing to the top of the pole.

### Step 2: Identify the Right Triangle
In this scenario, we can visualize a right triangle where:
- The height of the pole (AB) is the opposite side (10 m).
- The distance from the base of the pole to the point where the rope touches the ground (let's call it point C) is the adjacent side.
- The rope itself (AC) is the hypotenuse.

### Step 3: Use the Sine Function
We can use the sine function to relate the angle to the sides of the triangle. The sine of an angle in a right triangle is defined as:
\[ \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} \]
Here, \(\theta = 60^\circ\), the opposite side is the height of the pole (10 m), and the hypotenuse is the length of the rope (AC).

### Step 4: Set Up the Equation
Using the sine function:
\[ \sin(60^\circ) = \frac{10}{AC} \]
We know that \(\sin(60^\circ) = \frac{\sqrt{3}}{2}\). Therefore, we can substitute this value into the equation:
\[ \frac{\sqrt{3}}{2} = \frac{10}{AC} \]

### Step 5: Solve for AC
To find AC, we can rearrange the equation:
\[ AC = \frac{10 \times 2}{\sqrt{3}} = \frac{20}{\sqrt{3}} \text{ meters} \]

### Step 6: Rationalize the Denominator
To express AC in a more standard form, we can rationalize the denominator:
\[ AC = \frac{20}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{20\sqrt{3}}{3} \text{ meters} \]

### Final Answer for Part (i)
The distance covered by the artist in climbing to the top of the pole is:
\[ AC = \frac{20\sqrt{3}}{3} \text{ meters} \]

### Step 7: Identify the Mathematical Concept (Part ii)
The mathematical concept used in this problem is **Trigonometry**, specifically the use of the sine function in a right triangle.

### Step 8: State the Value (Part iii)
The value represented in this problem can be seen as **single-mindedness**, which emphasizes the importance of having a clear goal and determination to achieve it.