Calculating Tower Shadow Length Change using Trigonometry

This problem involves finding the change in the length of a tower's shadow as the sun's angle of elevation changes. We can solve this using the principles of trigonometry, specifically the tangent function, which relates the angle of elevation to the ratio of the height of the object (the tower) and the length of its shadow.

Understanding the Geometry and Angles of Elevation

Imagine a right-angled triangle formed by the tower (vertical side), the shadow on the ground (horizontal side), and the line of sight from the tip of the shadow to the top of the tower (hypotenuse). The angle of elevation of the sun is the angle between the horizontal shadow and the line of sight.

We have two scenarios:

1. When the angle of elevation is 60°. Let the length of the shadow be \(y\) meters.
2. When the angle of elevation is 45°. The shadow is \(x\) meters longer, so its length is \(y+x\) meters.

The height of the tower, \(h\), is given as \(5(3 + \sqrt{3})\) meters.

Applying the Tangent Function

The tangent of an angle in a right-angled triangle is defined as the ratio of the opposite side (height of the tower) to the adjacent side (length of the shadow).

For the first scenario (angle of elevation = 60°):

\[ \tan(60^\circ) = \frac{\text{Height of tower}}{\text{Length of shadow at 60°}} \]

\[ \tan(60^\circ) = \frac{h}{y} \]

We know that \( \tan(60^\circ) = \sqrt{3} \). So,

\[ \sqrt{3} = \frac{h}{y} \]

Rearranging to find \(y\):

\[ y = \frac{h}{\sqrt{3}} \quad \text{(Equation 1)} \]

For the second scenario (angle of elevation = 45°):

\[ \tan(45^\circ) = \frac{\text{Height of tower}}{\text{Length of shadow at 45°}} \]

\[ \tan(45^\circ) = \frac{h}{y+x} \]

We know that \( \tan(45^\circ) = 1 \). So,

\[ 1 = \frac{h}{y+x} \]

Rearranging to find \(y+x\):

\[ y+x = h \quad \text{(Equation 2)} \]

Solving for x, the Change in Shadow Length

Now we have a system of two equations with two unknowns, \(y\) and \(x\).

1. \( y = \frac{h}{\sqrt{3}} \)
2. \( y+x = h \)

Substitute Equation 1 into Equation 2:

\[ \frac{h}{\sqrt{3}} + x = h \]

Now, isolate \(x\):

\[ x = h - \frac{h}{\sqrt{3}} \]

Factor out \(h\):

\[ x = h \left( 1 - \frac{1}{\sqrt{3}} \right) \]

Simplify the term in the parenthesis:

\[ x = h \left( \frac{\sqrt{3}-1}{\sqrt{3}} \right) \]

We are given the height of the tower, \(h = 5(3 + \sqrt{3})\) meters. Substitute this value into the equation for \(x\):

\[ x = 5(3 + \sqrt{3}) \left( \frac{\sqrt{3}-1}{\sqrt{3}} \right) \]

\[ x = 5 \times \frac{(3 + \sqrt{3})(\sqrt{3}-1)}{\sqrt{3}} \]

Let's expand the numerator \((3 + \sqrt{3})(\sqrt{3}-1)\):

\[ (3 + \sqrt{3})(\sqrt{3}-1) = 3\sqrt{3} - 3 \times 1 + \sqrt{3} \times \sqrt{3} - \sqrt{3} \times 1 \]

\[ = 3\sqrt{3} - 3 + 3 - \sqrt{3} \]

\[ = (3\sqrt{3} - \sqrt{3}) + (-3 + 3) \]

\[ = 2\sqrt{3} + 0 \]

\[ = 2\sqrt{3} \]

Now substitute this back into the expression for \(x\):

\[ x = 5 \times \frac{2\sqrt{3}}{\sqrt{3}} \]

The \( \sqrt{3} \) in the numerator and the denominator cancel out:

\[ x = 5 \times 2 \]

\[ x = 10 \]

So, the value of \(x\) is 10 meters.

Final Answer

The increase in the length of the shadow, \(x\), when the angle of elevation changes from 60° to 45° is 10 meters.

Revision Table: Tower Shadow Problem

Additional Information: Angles and Trigonometric Ratios

Understanding standard angles and their trigonometric ratios is crucial for solving height and distance problems like this tower shadow calculation. The angles 60° and 45° are common angles with known tangent values:

• For 45°, the opposite and adjacent sides of the right triangle are equal. This means the height of the tower is equal to the length of the shadow when the angle of elevation is 45° (\( \tan(45^\circ) = 1 \)).
• For 60°, the tangent value is \( \sqrt{3} \), which is approximately 1.732. This means the shadow length is shorter than the tower height when the angle of elevation is 60° (\( \tan(60^\circ) = \sqrt{3} \)). A higher angle of elevation means a shorter shadow for the same height.

The problem demonstrates how a decrease in the angle of elevation (from 60° to 45°) leads to an increase in the shadow length, which makes sense geometrically.