Calculating Tower Height using Shadow Length and Sun Altitude

This problem involves using trigonometry, specifically the concept of angles of elevation (also known as the sun's altitude) and the properties of right-angled triangles. We have a vertical tower and its shadow on the level ground. As the sun's altitude changes, the length of the shadow also changes.

Let's define the terms and variables:

• Height of the tower: Let this be \(h\) meters. The tower is vertical, so it forms a right angle with the ground.
• Sun's Altitude: This is the angle of elevation from the tip of the shadow on the ground to the top of the tower.
• Shadow Length: The distance on the ground from the base of the tower to the tip of the shadow.

We are given two scenarios:

1. When the sun's altitude is 45°, let the shadow length be \(x\) meters.
2. When the sun's altitude changes to 30°, the shadow length increases by 10 meters. So, the new shadow length is \(x + 10\) meters.

We can represent these scenarios using two right-angled triangles. In each triangle, the height of the tower is the opposite side, and the shadow length is the adjacent side, relative to the angle of elevation (sun's altitude).

The trigonometric ratio that relates the opposite side and the adjacent side is the tangent function:

\(\tan(\text{angle}) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{\text{Height of Tower}}{\text{Shadow Length}}\)

Applying Tangent Function for Each Scenario

Scenario 1: Sun's Altitude = 45°

The triangle has height \(h\) and base (shadow length) \(x\). The angle is 45°.

\(\tan(45^\circ) = \frac{h}{x}\)

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

\(1 = \frac{h}{x}\)

This gives us our first equation:

\[h = x \quad \text{(Equation 1)}\]

Scenario 2: Sun's Altitude = 30°

The triangle has height \(h\) and base (shadow length) \(x + 10\). The angle is 30°.

\(\tan(30^\circ) = \frac{h}{x + 10}\)

We know that \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\). So,

\(\frac{1}{\sqrt{3}} = \frac{h}{x + 10}\)

This gives us our second equation:

\[\sqrt{3}h = x + 10 \quad \text{(Equation 2)}\]

Solving the System of Equations for Tower Height

Now we have two equations with two variables (\(h\) and \(x\)):

1) \(h = x\)

2) \(\sqrt{3}h = x + 10\)

We can substitute Equation 1 into Equation 2. Since \(x = h\), replace \(x\) with \(h\) in Equation 2:

\(\sqrt{3}h = h + 10\)

Now, we need to solve for \(h\). Gather the terms involving \(h\) on one side:

\(\sqrt{3}h - h = 10\)

Factor out \(h\) from the terms on the left side:

\(h(\sqrt{3} - 1) = 10\)

To isolate \(h\), divide both sides by \((\sqrt{3} - 1)\):

\(h = \frac{10}{\sqrt{3} - 1}\)

Rationalizing the Denominator

It is standard practice to rationalize the denominator when it contains a square root expression like \((\sqrt{3} - 1)\). We multiply the numerator and the denominator by the conjugate of the denominator, which is \((\sqrt{3} + 1)\):

\(h = \frac{10}{\sqrt{3} - 1} \times \frac{\sqrt{3} + 1}{\sqrt{3} + 1}\)

Using the difference of squares formula, \((a - b)(a + b) = a^2 - b^2\), for the denominator:

\((\sqrt{3} - 1)(\sqrt{3} + 1) = (\sqrt{3})^2 - (1)^2 = 3 - 1 = 2\)

So, the expression for \(h\) becomes:

\(h = \frac{10(\sqrt{3} + 1)}{2}\)

Now, simplify by dividing 10 by 2:

\(h = 5(\sqrt{3} + 1)\)

The height of the tower is \(5(\sqrt{3} + 1)\) meters.

Let's check the options provided:

• 10(\(\sqrt{3}\) + 1) m
• 5\(\sqrt{3}\) m
• 5(\(\sqrt{3}\) + 1) m
• 10\(\sqrt{3}\) m

Our calculated height \(h = 5(\sqrt{3} + 1)\) m matches the third option.

Revision Table: Key Trigonometric Values

Additional Information: Angles of Elevation and Depression

In trigonometry, the angle of elevation is the angle between the horizontal line of sight and the line of sight upwards to an object. In this problem, the sun's altitude is the angle of elevation from the ground (tip of the shadow) up to the top of the tower. As the sun gets higher in the sky, its altitude increases, and the shadow of an object gets shorter. Conversely, as the sun gets lower (altitude decreases), the shadow gets longer, which is consistent with the problem statement where the shadow length increased when the altitude changed from 45° to 30°.

The angle of depression is similar but is the angle between the horizontal line of sight and the line of sight downwards to an object.