Question

From the top of a tower, the angles of depression of the top and bottom of a building are \(30^{\circ}\) and \(45^{\circ}\) respectively. If the height of the building is \(40 \mathrm{~m}\), the height of the tower is (a) \(20(3+\sqrt{3})\) (b) \(20(\sqrt{3}+1)\) (c) \(40(3+\sqrt{3})\) (d) \(40(\sqrt{3}+1)\)

Short answer

Using the given angles of depression and the height of the building, we can set up right triangles and use the tangent function to create equations involving the distances and heights. By solving these equations and finding the relationship between the height difference and the total height, we can determine the height of the tower. The height of the tower is \(40(\sqrt{3} + 1)\), which simplifies to \(20(\sqrt{3} + 1)\).

Step-by-step solution

Understand the problem and draw a diagram

Read the problem carefully and draw a diagram to represent the given information. Draw the tower and the building and mark the given angles of depression and height. Label the points and distances as needed.

Set up right triangles

From the diagram, we can see that we have two right triangles. Triangle 1 is formed with the top of the tower, the top of the building, and the bottom of the building. Triangle 2 is formed with the top of the tower, the bottom of the tower, and the bottom of the building. We will use these triangles to find the height of the tower.

Using the tangent function for Triangle 1

Since the angle of depression of the top of the building is given to be \(30^{\circ}\), the angle of elevation from the top of the building to the top of the tower is also \(30^{\circ}\). Let the height difference between the top of the building and the top of the tower be \(h_1\). Then we can use the tangent function to find the horizontal distance \(d\) between the building and the tower. We have \(\tan(30^{\circ}) = \frac{h_1}{d}\).

Using the tangent function for Triangle 2

For Triangle 2, the angle of elevation from the bottom of the building to the top of the tower is given to be \(45^{\circ}\). Let the total height of the tower be \(h\). Then we can use the tangent function to find the horizontal distance \(d\) between the building and the tower. We have \(\tan(45^{\circ}) = \frac{h}{d}\).

Solve the equations for \(d\) and \(h\)

Express \(d\) in terms of \(h_1\) from the equation involving Triangle 1: \(d = \frac{h_1}{\tan(30^{\circ})}\). Now, express \(d\) in terms of \(h\) from the equation involving Triangle 2: \(d = \frac{h}{\tan(45^{\circ})}\). Equate the two expressions for \(d\): \(\frac{h_1}{\tan(30^{\circ})} = \frac{h}{\tan(45^{\circ})}\).

Find the relationship between \(h_1\) and \(h\)

Substitute the values of \(\tan(30^{\circ})\) and \(\tan(45^{\circ})\) to get: \(\frac{h_1}{\frac{1}{\sqrt{3}}} = \frac{h}{1}\). Simplify this equation to find the relationship between \(h_1\) and \(h\): \(h_1 = h(\frac{1}{\sqrt{3}})\).

Find the height of the tower \(h\)

We know that the height of the building is \(40 \mathrm{~m}\), and the height difference between the top of the building and the top of the tower is \(h_1\). Therefore, we have \(h = 40 + h_1\). Substitute the expression for \(h_1\) found in step 6: \(h = 40 + h(\frac{1}{\sqrt{3}})\). Solve the equation for \(h\): \(h = 40(\sqrt{3} + 1)\). The height of the tower is \(40(\sqrt{3} + 1)\), so the correct answer is (b) \(20(\sqrt{3} + 1)\).

Key concepts

Angles of Depression

When learning about angles of depression, it's important to understand that they are the angles formed by the line of sight when an observer looks downward. Imagine standing on top of a tower and staring at a building below; the angle measured from the horizontal line of sight to the line directed towards the building's top or base is the angle of depression. This concept is crucial in various fields such as architecture, aviation, and even simple tasks like measuring the height of structures from a distance.

• Angle of depression is equal to the angle of elevation from the object's top or bottom point towards the observer's eye level, forming congruent angles in geometry.
• In solving mathematical problems involving angles of depression, always use a horizontal line from the observer's view as a reference point.
• Draw a clear diagram to mark angles and distances for a better visual representation of the problem.

Understanding angles of depression helps in accurately assessing distances and heights in real-world scenarios.

Tangent Function

The tangent function in trigonometry is a fundamental tool used to solve problems involving right triangles and angles. It is especially handy when dealing with angles of depression and elevation. The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side.

• Mathematically, the tangent of an angle \( \theta \) is given by \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \).
• This function is pivotal when there is a need to find distances or heights, which are not directly measurable.
• For example, given an angle of \(30^\circ\) with an opposite side (height difference) and need for the horizontal distance, one can use \(\tan(30^\circ) = \frac{h}{d} \).

Knowing how to apply the tangent function makes solving real-world distance and angle problems much easier and accurate.

Right Triangles

The understanding of right triangles is essential when working with trigonometric problems, especially those involving tangent functions and angles of depression. A right triangle has one angle exactly equal to \(90^\circ\), with the other two angles being acute and adding up to \(90^\circ\).

• Key sides in right triangles include: the hypotenuse (the longest side opposite the right angle), and the two legs – one adjacent to the angle of interest and one opposite it.
• Right triangles often appear in problems involving elevation and depression, serving as the geometric basis that aids in calculations.
• Understanding side relations and angle properties is key to applying trigonometric functions accurately in such triangles.

The concept of right triangles is useful because it provides a simple framework for breaking down complex trigonometric problems into manageable parts.

Problem Solving in Mathematics

Problem solving in mathematics is not just about arriving at the correct answer, but it also involves understanding the process and reasoning logically through each step. Tackling a problem like the one involving angles of depression requires a series of logical steps:

• First, carefully read the problem to grasp all given elements, ensuring all details are noted.
• Next, illustrate the situation with a diagram. This visual aid proves indispensable in understanding the relationships between different elements.
• Identify known and unknown quantities, and choose appropriate mathematical methods – in this case, the tangent function for angles of depression.
• Solve equations step-by-step, consistently checking that each step follows logically from the last.

Effective problem solving equips students with the skills to tackle a variety of mathematical challenges, building confidence and proficiency over time.