A person of height 6ft .wants to pluck a fruit which is on a `(26)/(3)` ft. high tree .If the person is standing `(8)/(sqrt(3))` ft . Away from the base of the tree ,then at what angle should he throw a stone so that it hits the fruit ?

Options

A. `75^(@)`
B. `30^(@)`
C. `45^(@)`
D. `60^(@)`

Answer

To solve the problem, we need to find the angle at which the person should throw the stone to hit the fruit on the tree. We can break down the solution into several steps:

### Step 1: Understand the heights involved
- The height of the tree where the fruit is located is \( \frac{26}{3} \) feet.
- The height of the person is 6 feet.

### Step 2: Calculate the height difference
- The height from which the stone needs to be thrown (the height of the fruit above the person's head) is calculated as follows:
\[ \text{Height difference} = \text{Height of tree} - \text{Height of person} = \frac{26}{3} - 6 \]
To perform the subtraction, convert 6 into a fraction with a denominator of 3:
\[ 6 = \frac{18}{3} \]
Thus,
\[ \text{Height difference} = \frac{26}{3} - \frac{18}{3} = \frac{8}{3} \text{ feet} \]

### Step 3: Determine the horizontal distance
- The person is standing \( \frac{8}{\sqrt{3}} \) feet away from the base of the tree.

### Step 4: Set up the tangent function
- The angle \( \theta \) at which the stone should be thrown can be found using the tangent function, which is defined as:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
In our case:
\[ \tan(\theta) = \frac{\text{Height difference}}{\text{Horizontal distance}} = \frac{\frac{8}{3}}{\frac{8}{\sqrt{3}}} \]

### Step 5: Simplify the tangent expression
- Simplifying the fraction:
\[ \tan(\theta) = \frac{8}{3} \cdot \frac{\sqrt{3}}{8} = \frac{\sqrt{3}}{3} \]
Thus,
\[ \tan(\theta) = \frac{1}{\sqrt{3}} \]

### Step 6: Find the angle
- The angle \( \theta \) whose tangent is \( \frac{1}{\sqrt{3}} \) is:
\[ \theta = 30^\circ \]

### Conclusion
- Therefore, the angle at which the person should throw the stone to hit the fruit is \( 30^\circ \).