Understanding the Tower Height Problem

This question asks us to find the height of a tower using information about angles of elevation observed from two points on a straight line passing through the foot of the tower. We are given the distances of these two observation points from the base of the tower and the corresponding angles measured upwards to the top of the tower.

Defining Variables and Setting up the Scenario

Let's define the variables we will use:

• Let \(h\) represent the height of the tower.
• Let the foot of the tower be denoted by point F, and the top of the tower by point T.
• Let the two points on the ground, on a line through F, be A and B.
• The distance from F to A is \(d_1 = 49\) m.
• The distance from F to B is \(d_2 = 36\) m.
• The angle of elevation from point A to the top of the tower T is \(\theta_1 = 43^\circ\).
• The angle of elevation from point B to the top of the tower T is \(\theta_2 = 47^\circ\).

Since the angle of elevation increases as an observer moves closer to the base of the tower, the smaller distance (36 m) corresponds to the larger angle (\(47^\circ\)), and the larger distance (49 m) corresponds to the smaller angle (\(43^\circ\)). Assuming the points A and B are on the same side of the tower's foot, we have two right-angled triangles, \(\triangle AFT\) and \(\triangle BFT\).

Using Trigonometry (Tangent Ratio)

In a right-angled triangle, the tangent of an angle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. In our setup:

For \(\triangle AFT\), the side opposite to \(\theta_1\) is the height of the tower \(h\), and the side adjacent is the distance \(d_1\). So:

\[ \tan(\theta_1) = \frac{h}{d_1} \]

\[ \tan(43^\circ) = \frac{h}{49} \quad \text{(Equation 1)} \]

For \(\triangle BFT\), the side opposite to \(\theta_2\) is the height of the tower \(h\), and the side adjacent is the distance \(d_2\). So:

\[ \tan(\theta_2) = \frac{h}{d_2} \]

\[ \tan(47^\circ) = \frac{h}{36} \quad \text{(Equation 2)} \]

Identifying Complementary Angles

Let's look at the angles given: \(43^\circ\) and \(47^\circ\). We notice that their sum is \(43^\circ + 47^\circ = 90^\circ\). This means the two angles of elevation are complementary.

An important property of complementary angles is that the tangent of one angle is equal to the cotangent of the other. Since \(\cot(\theta) = \frac{1}{\tan(\theta)}\), if \(\theta_1 + \theta_2 = 90^\circ\), then \(\tan(\theta_2) = \tan(90^\circ - \theta_1) = \cot(\theta_1) = \frac{1}{\tan(\theta_1)}\).

Deriving Height using Complementary Angles

From Equation 1, we can express \(\tan(\theta_1)\) as \(\tan(43^\circ) = \frac{h}{49}\).

From Equation 2, we have \(\tan(47^\circ) = \frac{h}{36}\). Since \(47^\circ = 90^\circ - 43^\circ\), we can write:

\[ \tan(47^\circ) = \cot(43^\circ) = \frac{1}{\tan(43^\circ)} \]

Substituting this into Equation 2:

\[ \frac{1}{\tan(43^\circ)} = \frac{h}{36} \]

Now, substitute the expression for \(\tan(43^\circ)\) from Equation 1 (\(\tan(43^\circ) = \frac{h}{49}\)) into the equation above:

\[ \frac{1}{\left(\frac{h}{49}\right)} = \frac{h}{36} \]

\[ \frac{49}{h} = \frac{h}{36} \]

Cross-multiplying the terms:

\[ h \times h = 49 \times 36 \]

\[ h^2 = 49 \times 36 \]

To find \(h\), take the square root of both sides:

\[ h = \sqrt{49 \times 36} \]

This formula \(h = \sqrt{d_1 d_2}\) is a specific result for problems where the angles of elevation from two points on a line through the base are complementary, and the points are on the same side of the tower.

Calculating the Final Tower Height

Now, we perform the calculation using the distances \(d_1 = 49\) m and \(d_2 = 36\) m:

\[ h = \sqrt{49 \times 36} \]

We can find the square roots of 49 and 36 separately:

\[ \sqrt{49} = 7 \]

\[ \sqrt{36} = 6 \]

So, the height \(h\) is:

\[ h = 7 \times 6 \]

\[ h = 42 \text{ m} \]

The height of the tower is 42 meters.

Revision Table: Key Concepts Recap

Additional Information: Solving Height and Distance Problems

Height and distance problems are classic applications of trigonometry. They typically involve right-angled triangles formed by a vertical object (like a tower, building, or tree), the ground, and the line of sight from an observer to the top or bottom of the object.

Steps often involved in solving such problems:

1. Draw a clear diagram representing the situation.
2. Identify the right-angled triangles formed.
3. Label the known distances, angles, and the unknown quantity (the height of the tower in this case).
4. Use appropriate trigonometric ratios (sine, cosine, or tangent) based on the sides and angles involved in the right triangle(s). The tangent function is very common for problems involving height and horizontal distance.
5. Set up trigonometric equation(s).
6. Solve the equation(s) to find the unknown quantity. If there are two points of observation, you might get a system of two equations to solve simultaneously, or, as in this case with complementary angles, a simpler relationship might emerge.

Understanding the relationships between trigonometric ratios for special angles and angle properties (like complementary angles) can often simplify calculations.