Question

The shadow of a tower standing on a level ground is found to be 40 m longer when the Sun’s altitude is 30° than when it was 60°. Find the height of the tower and the length of the original shadow. (use \(\sqrt{ 3}\) = 1.73)

Answer 1

Step 1: Understanding the problem:
We are given a tower standing on level ground. The shadow of the tower is found to be 40 m longer when the Sun’s altitude is 30° than when the Sun’s altitude is 60°. We need to find the height of the tower and the length of the original shadow.

Let the height of the tower be \( h \, \text{m} \), and let the length of the shadow when the Sun’s altitude is 60° be \( x \, \text{m} \).
We will use the trigonometric tangent function, which is defined as the ratio of the height of the object (tower) to the length of its shadow:
\[ \tan \theta = \frac{\text{height of the tower}}{\text{length of the shadow}} \] Thus, for the two situations: 
Step 2: Set up equations using the tangent function:
1. When the Sun’s altitude is 60°:
\[ \tan 60^\circ = \frac{h}{x} \] Since \( \tan 60^\circ = \sqrt{3} \), we have the equation:
\[ \sqrt{3} = \frac{h}{x} \quad \Rightarrow \quad h = \sqrt{3} \cdot x \] 2. When the Sun’s altitude is 30°:
\[ \tan 30^\circ = \frac{h}{x + 40} \] Since \( \tan 30^\circ = \frac{1}{\sqrt{3}} \), we have the equation:
\[ \frac{1}{\sqrt{3}} = \frac{h}{x + 40} \quad \Rightarrow \quad h = \frac{x + 40}{\sqrt{3}} \] Step 3: Solve the system of equations:
Now, we have two equations for \( h \):
1. \( h = \sqrt{3} \cdot x \)
2. \( h = \frac{x + 40}{\sqrt{3}} \)
Equating the two expressions for \( h \):
\[ \sqrt{3} \cdot x = \frac{x + 40}{\sqrt{3}} \] Multiply both sides by \( \sqrt{3} \) to eliminate the denominator:
\[ 3x = x + 40 \] Now, solve for \( x \):
\[ 3x - x = 40 \quad \Rightarrow \quad 2x = 40 \quad \Rightarrow \quad x = 20 \] Thus, the length of the shadow when the Sun’s altitude is 60° is \( x = 20 \, \text{m} \).

Step 4: Find the height of the tower:
Substitute \( x = 20 \) into the equation \( h = \sqrt{3} \cdot x \):
\[ h = \sqrt{3} \cdot 20 = 1.73 \cdot 20 = 34.6 \, \text{m} \] Conclusion:
The height of the tower is approximately \( 34.6 \, \text{m} \), and the length of the original shadow is \( 20 \, \text{m} \).