If the angle of elevation of the sum at the given time is 30° then find the length of the shadow cast by the tower of 150 feet height at that time.

tan 30 = 150/x

To find the length of the shadow cast by the tower, we can use the tangent function.

Step 1: First, let's identify the known values:
- The height of the tower is 150 feet.
- The angle of elevation of the sun is 30°.

Step 2: Now, we can use the tangent function to find the length of the shadow.
The tangent of an angle is equal to the opposite side divided by the adjacent side.

tan(30°) = opposite (shadow length) / adjacent (tower height)

Step 3: Substitute the known values into the equation and solve for the shadow length:
tan(30°) = shadow length / 150 feet

We can rearrange the equation to solve for the shadow length:
shadow length = tan(30°) * 150 feet

Step 4: Calculate the shadow length:
tan(30°) is approximately 0.5774

shadow length ≈ 0.5774 * 150 feet
shadow length ≈ 86.61 feet

Therefore, the length of the shadow cast by the tower is approximately 86.61 feet when the angle of elevation of the sun is 30°.

To find the length of the shadow cast by the tower, we can use trigonometry.

Let's assume the length of the shadow to be 'x'.

Since the angle of elevation of the sun is 30°, the angle of depression from the top of the tower to the tip of its shadow would also be 30°.

In this scenario, we have a right triangle formed by the tower, its shadow, and the line from the top of the tower to the tip of its shadow. The height of the tower (opposite side of the angle) is 150 feet, and we need to find the length of the shadow (adjacent side of the angle), which is 'x'.

Using the tangent function, we have:

tan(30°) = opposite/adjacent
tan(30°) = 150/x

We can solve this equation to find the value of 'x':

x = 150 / tan(30°)

To evaluate this expression, we need to determine the value of tan(30°). The tangent of 30° is equal to the square root of 3 divided by 3 (√3/3). Therefore:

x = 150 / (√3/3)

To simplify this expression further, we multiply both the numerator and denominator by the conjugate of (√3/3), which is (√3/3):

x = 150 * (3/√3) = 150 * (3√3/3) = 150√3

Therefore, the length of the shadow cast by the tower at the given time is 150√3 feet.