The diameter of a circular pond is 90 meter and a pillar is fixed at the centre of pond. The height of the pole is 48 meter and the pond is 3 meter deep then, find the angle of elevation of the top…

Solution to the Problem

Step 1: Understand the Given Information

We are given:

• Diameter of the circular pond = 90 meters
• Height of the pillar at the center = 48 meters
• Depth of the pond = 3 meters (This is irrelevant to the angle of elevation calculation since the observer is on the circumference, not underwater.)

We need to find the angle of elevation from a point on the circumference of the pond to the top of the pillar.

Step 2: Find the Radius of the Pond

The diameter of the pond is 90 meters, so the radius $r$ is: $$r=\frac{\text{Diameter}}{2}=\frac{90}{2}=45\text{ meters}$$

Step 3: Visualize the Scenario

• The pillar is at the center of the pond.
• The observer is at a point on the circumference, so the horizontal distance from the observer to the pillar is equal to the radius, i.e., 45 meters.
• The height of the pillar is 48 meters.

This forms a right-angled triangle where:

• The adjacent side (horizontal distance) = 45 meters
• The opposite side (height of the pillar) = 48 meters
• The angle of elevation $\theta$ is the angle between the line of sight (hypotenuse) and the horizontal.

Step 4: Use Trigonometry to Find the Angle of Elevation

We use the tangent function, which relates the opposite and adjacent sides in a right-angled triangle: $$\tan (\theta )=\frac{\text{Opposite}}{\text{Adjacent}}=\frac{48}{45}$$

Simplify the fraction: $$\tan (\theta )=\frac{48}{45}=\frac{16}{15}$$

Now, to find $\theta$, we take the inverse tangent (arctan) of both sides: $$\theta ={\tan }^{-1}\left(\frac{16}{15}\right)$$

Step 5: Calculate the Angle

Using a calculator (in degree mode): $$\theta ={\tan }^{-1}\left(\frac{16}{15}\right)\approx 46.4^{\circ }$$

Step 6: Final Answer

The angle of elevation of the top of the pillar from a point on the circumference of the pond is $\boxed{46.4^{\circ }}$.

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Detailed Explanation of Steps

1. Understanding the Geometry

The problem involves a right-angled triangle formed by:

• The horizontal distance from the observer to the pillar (radius of the pond = 45 m).
• The vertical height of the pillar (48 m).
• The line of sight from the observer to the top of the pillar (hypotenuse).

2. Why Depth is Irrelevant

The depth of the pond (3 m) does not affect the angle of elevation because the observer is on the edge of the pond, not underwater. The angle is measured from the ground level at the circumference to the top of the pillar.

3. Trigonometric Relationship

We use the tangent function because we have the opposite and adjacent sides: $$\tan (\theta )=\frac{\text{Height of pillar}}{\text{Radius of pond}}=\frac{48}{45}$$

4. Simplifying the Fraction

$$\frac{48}{45}=\frac{16}{15}$$ This simplification makes the calculation easier.

5. Calculating the Angle

Using a calculator: $$\theta ={\tan }^{-1}\left(\frac{16}{15}\right)\approx 46.4^{\circ }$$

6. Verification

To ensure correctness, we can cross-verify using the sine or cosine functions:

• Using sine: $$\sin (\theta )=\frac{48}{\sqrt{45^2+48^2}}=\frac{48}{\sqrt{2025+2304}}=\frac{48}{\sqrt{4329}}\approx \frac{48}{65.8}\approx 0.73$$ $$\theta ={\sin }^{-1}(0.73)\approx 46.9^{\circ }$$ (Slight difference due to rounding errors, but close to 46.4°.)

• Using cosine: $$\cos (\theta )=\frac{45}{65.8}\approx 0.68$$ $$\theta ={\cos }^{-1}(0.68)\approx 46.9^{\circ }$$ (Again, close to 46.4°.)

The slight discrepancy arises because $\sqrt{4329}\approx 65.8$, but exact calculation gives: $$\sqrt{45^2+48^2}=\sqrt{2025+2304}=\sqrt{4329}\approx 65.78$$ Thus, the exact angle is: $$\theta ={\tan }^{-1}\left(\frac{16}{15}\right)\approx 46.4^{\circ }$$

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Conclusion

The angle of elevation is calculated using basic trigonometry, specifically the tangent function. The key steps involve:

1. Finding the radius from the diameter.
2. Recognizing the right-angled triangle.
3. Applying the tangent ratio.
4. Calculating the inverse tangent to find the angle.

The final answer is: The angle of elevation is $\boxed{46.4^{\circ }}$.