Understanding the Problem: Tower, Shadow, and Angle of Elevation

This question involves a classic application of trigonometry, specifically dealing with angles of elevation. We are given a tower of a certain height. When the sun's angle of elevation changes, the length of the tower's shadow also changes. We are provided with the tower's height, the initial angle of elevation, the increase in shadow length due to a change in the angle, and we need to determine the range of the new angle of elevation.

Let's define the key components:

• Height of the tower: \(h\)
• Initial angle of elevation of the sun: \(\alpha = 60^\circ\)
• Initial length of the shadow: \(y\)
• New angle of elevation of the sun: \(\theta\)
• New length of the shadow: \(y'\)
• Increase in shadow length: \(x\) (so, \(y' = y + x\))

We are given that the height of the tower is \(h = \sqrt{3}x\) metres.

Applying Trigonometry to the Tower and Shadow

The relationship between the height of the tower, the length of its shadow, and the angle of elevation of the sun is given by the tangent function in a right-angled triangle:

\(\tan(\text{angle of elevation}) = \frac{\text{Height of the tower}}{\text{Length of the shadow}}\)

Scenario 1: Initial Angle of Elevation (\(60^\circ\))

When the angle of elevation is \(60^\circ\), the shadow length is \(y\). Using the tangent function:

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

We know \(h = \sqrt{3}x\) and \(\tan(60^\circ) = \sqrt{3}\). Substituting these values:

\(\sqrt{3} = \frac{\sqrt{3}x}{y}\)

Now, we can solve for the initial shadow length \(y\):

\(y \times \sqrt{3} = \sqrt{3}x\)

\(y = \frac{\sqrt{3}x}{\sqrt{3}}\)

\(y = x\)

So, the initial shadow length is \(x\) metres.

Scenario 2: New Angle of Elevation (\(\theta\))

When the angle of elevation changes to \(\theta\), the shadow becomes \(x\) metre longer. This means the new shadow length \(y'\) is \(y + x\). Since we found \(y = x\), the new shadow length is:

\(y' = y + x = x + x = 2x\)

Now, using the tangent function for the new angle \(\theta\) and the new shadow length \(2x\), with the tower height \(h = \sqrt{3}x\):

\(\tan(\theta) = \frac{h}{y'}\)

\(\tan(\theta) = \frac{\sqrt{3}x}{2x}\)

We can cancel out \(x\) from the numerator and denominator (assuming \(x \neq 0\), which must be true since the shadow becomes \(x\) metre longer):

\(\tan(\theta) = \frac{\sqrt{3}}{2}\)

Determining the Range of \(\theta\)

We have found that \(\tan(\theta) = \frac{\sqrt{3}}{2}\). To find the range of \(\theta\), we need to compare this value with the tangent of known angles, particularly those mentioned in the options.

We know the values of tangent for standard angles:

• \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\)
• \(\tan(45^\circ) = 1\)
• \(\tan(60^\circ) = \sqrt{3}\)

Let's approximate the value we found:

\(\tan(\theta) = \frac{\sqrt{3}}{2} \approx \frac{1.732}{2} \approx 0.866\)

Now compare this value with the tangent of \(30^\circ\) and \(45^\circ\):

• \(\tan(30^\circ) = \frac{1}{\sqrt{3}} \approx \frac{1}{1.732} \approx 0.577\)
• \(\tan(45^\circ) = 1\)

We can see that \(0.577 < 0.866 < 1\). Therefore, \(\tan(30^\circ) < \tan(\theta) < \tan(45^\circ)\).

Since the tangent function is strictly increasing for angles between \(0^\circ\) and \(90^\circ\), this inequality holds for the angles themselves:

\(30^\circ < \theta < 45^\circ\)

This range for \(\theta\) matches one of the given options.

Summary of Steps

1. Identify the given information: height of tower, initial angle, increase in shadow length.
2. Set up trigonometric equations (using tangent) for the initial and new scenarios.
3. Solve for the initial shadow length using the initial angle and height.
4. Calculate the new shadow length using the initial shadow length and the given increase.
5. Use the new shadow length and height to find the value of \(\tan(\theta)\).
6. Compare the value of \(\tan(\theta)\) with known tangent values to determine the range of \(\theta\).

Based on our calculation, the angle \(\theta\) falls within the range of \(30^\circ\) and \(45^\circ\).

Revision Table: Key Trigonometric Values

Comparing the values, we see that \(0.577 < 0.866 < 1\), confirming that \(30^\circ < \theta < 45^\circ\).

Additional Information: Angles of Elevation and Depression

The angle of elevation is the angle formed by the horizontal line of sight and the line of sight upwards to an object. In this problem, the object is the sun, and the line of sight is from the tip of the shadow (on the ground) to the top of the tower.

As the sun gets lower in the sky (its angle of elevation decreases), the shadow cast by an object gets longer. Conversely, as the sun gets higher (its angle of elevation increases), the shadow gets shorter. This is consistent with our findings: the angle changed from \(60^\circ\) to a smaller angle \(\theta\), and the shadow length increased.

The angle of depression is similar, but it's the angle formed by the horizontal line of sight and the line of sight downwards to an object.

Problems involving heights and distances often use the tangent function because it directly relates the opposite side (height) and the adjacent side (distance or shadow length) in a right-angled triangle formed by the object, the ground, and the line of sight.