Deducing the Angle of Elevation of the Sun with a Simple Shadow Pole

Understanding the angle of elevation of the sun is crucial for various purposes, including solar energy harvesting, environmental studies, and astronomy. In this article, we’ll explore how to find the sun's angle of elevation using a simple shadow pole measurement. The problem involves a 7-meter pole with a shadow that is half its length. Using trigonometric principles, we will deduce the angle of elevation of the sun.

Understanding the Problem

The problem states that the shadow of a 7-meter high pole is exactly half the height of the pole. This means the shadow is 3.5 meters long. We need to find the angle of elevation of the sun, which forms a right-angled triangle with the pole and its shadow.

Applying Trigonometry to Find the Angle of Elevation

To solve this problem, we can use the tangent function, which is defined as the ratio of the opposite side to the adjacent side in a right-angled triangle. In this case, the height of the pole (7 meters) is the opposite side, and the length of the shadow (3.5 meters) is the adjacent side.

Step-by-Step Solution

Identify the sides of the triangle:
Height of the pole (opposite): ( h 7 ) meters Length of the shadow (adjacent): ( s frac{1}{2} times h frac{1}{2} times 7 3.5 ) meters Calculate the tangent of the angle of elevation:

(tan theta frac{h}{s} frac{7}{3.5} 2)

Find the angle using the arctangent function:

(theta arctan(2) approx 63.43^circ)

Round to the nearest degree:

(theta approx 63^circ)

Alternative Approach Using L h/tan(α)

An alternative method to find the angle of elevation is by using the formula ( L frac{h}{tan alpha} ), where ( L ) is the length of the shadow, ( h ) is the height of the pole, and ( alpha ) is the angle of the sun.

Given:

Pole height (( h )) 7 meters Shadow length (( L )) 3.5 meters

Solving for ( tan alpha ):

(tan alpha frac{h}{L} frac{7}{3.5} 2)

(alpha arctan(2) approx 63.43^circ)

ArcTan(2) results in approximately 63.43 degrees, which, when rounded to the nearest degree, equals 63 degrees.

Using the Pythagorean Theorem

Another approach is to use the Pythagorean theorem to find the hypotenuse of the right-angled triangle formed by the pole, its shadow, and the sun’s ray.

The hypotenuse ( H ) can be calculated as:

( H sqrt{7^2 3.5^2} sqrt{49 12.25} sqrt{61.25} approx 7.83 ) meters

To find the angle of elevation, we can use the sine or tangent function:

(sin theta frac{7}{7.83} approx 0.89) (theta arcsin(0.89) approx 62.9^circ)

(tan theta frac{7}{3.5} 2) (theta arctan(2) approx 63.4^circ)

Rounding to the nearest degree, the angle of elevation of the sun is 63 degrees.

Conclusion

The angle of elevation of the sun, given that the shadow is half the height of the pole (7 meters high), is approximately 63 degrees. This can be determined using trigonometric principles such as the tangent, sine, and Pythagorean theorem. Understanding these concepts can provide valuable insights into solar behavior and its applications in various fields.