Chapter 9 Some Applications of Trigonometry

Case Study 1: The Height of a Building

Case Description:
In a city, a surveyor is tasked with determining the height of a newly constructed building. He stands at a distance of 50 meters from the base of the building. Using a clinometer, he measures the angle of elevation to the top of the building, which is found to be 60 degrees. The surveyor knows the formula for height $h$ of the building, which is given by $h = d \cdot \tan(\theta)$, where $d$ is the distance from the building and $\theta$ is the angle of elevation.

After calculating the height, the surveyor prepares a report to provide valuable data for urban planning.

MCQs:

1. What is the distance $d$ from the base of the building where the surveyor stands?

   • A) 40 meters
   • B) 50 meters
   • C) 60 meters
   • D) 70 meters

2. What is the angle of elevation measured by the surveyor?

   • A) 45 degrees
   • B) 60 degrees
   • C) 30 degrees
   • D) 90 degrees

3. Using the formula $h = d \cdot \tan(\theta)$, what is the height $h$ of the building?

   • A) 50 meters
   • B) $50\sqrt{3}$ meters
   • C) $25\sqrt{3}$ meters
   • D) 100 meters

4. If the angle of elevation were to decrease to 45 degrees while maintaining the same distance, what would be the new height of the building?

   • A) 50 meters
   • B) 40 meters
   • C) 30 meters
   • D) 25 meters

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Case Study 2: The Angle of Depression

Case Description:
A person is standing on a cliff that is 100 meters high. Looking down at a boat in the water, he measures the angle of depression to the boat as 30 degrees. The angle of depression is the angle formed by the line of sight down to the boat and the horizontal line from the observer's eye. To find the horizontal distance from the base of the cliff to the boat, the person uses trigonometric relationships. This information helps in navigating the boat safely to the shore.

MCQs:

1. What is the height of the cliff?

   • A) 80 meters
   • B) 90 meters
   • C) 100 meters
   • D) 110 meters

2. What is the angle of depression measured by the person?

   • A) 45 degrees
   • B) 30 degrees
   • C) 60 degrees
   • D) 90 degrees

3. Which trigonometric function can be used to find the horizontal distance $d$ to the boat from the cliff?

   • A) $\sin$
   • B) $\tan$
   • C) $\cos$
   • D) All of the above

4. If the distance $d$ is calculated, what is its value if the angle of depression is 30 degrees?

   • A) 50 meters
   • B) 80 meters
   • C) 100 meters
   • D) 173.21 meters

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Case Study 3: A Ramp for Loading Goods

Case Description:
A loading dock uses a ramp to help transport goods from the ground to the height of the dock, which is 2 meters above the ground. The angle of elevation of the ramp is measured to be 45 degrees. The workers need to ensure that the ramp is not too steep for the wheeled carts carrying heavy goods. By using trigonometry, they can calculate the length of the ramp needed.

The ramp’s length can be determined using the relationship $\text{Length} = \frac{\text{Height}}{\sin(\theta)}$.

MCQs:

1. What is the height of the loading dock?

   • A) 1 meter
   • B) 2 meters
   • C) 3 meters
   • D) 4 meters

2. What is the angle of elevation of the ramp?

   • A) 30 degrees
   • B) 45 degrees
   • C) 60 degrees
   • D) 90 degrees

3. Using the formula, what is the length of the ramp?

   • A) 2 meters
   • B) $2\sqrt{2}$ meters
   • C) 4 meters
   • D) 5 meters

4. If the height of the dock increases to 3 meters while maintaining the angle of elevation, what will be the new length of the ramp?

   • A) $3\sqrt{2}$ meters
   • B) 3 meters
   • C) 4 meters
   • D) $\sqrt{3}$ meters

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Case Study 4: The Telescope Observation

Case Description:
An astronomer is using a telescope to observe a star. The telescope is positioned on the ground, and the angle of elevation to the star is measured to be 60 degrees. If the distance from the base of the telescope to the point directly below the star is 100 meters, the astronomer can calculate how high the star is above the ground using trigonometric functions. This data is crucial for understanding the star's position relative to Earth.

MCQs:

1. What is the angle of elevation to the star?

   • A) 30 degrees
   • B) 45 degrees
   • C) 60 degrees
   • D) 90 degrees

2. What is the horizontal distance from the base of the telescope to the point directly below the star?

   • A) 50 meters
   • B) 75 meters
   • C) 100 meters
   • D) 150 meters

3. Which trigonometric ratio can be used to calculate the height $h$ of the star?

   • A) $\sin \theta$
   • B) $\cos \theta$
   • C) $\tan \theta$
   • D) All of the above

4. What is the height of the star if the distance is 100 meters and the angle of elevation is 60 degrees?

   • A) 100 meters
   • B) 150 meters
   • C) $100\sqrt{3}$ meters
   • D) 50 meters

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Case Study 5: Navigating by the North Star

Case Description:
A sailor is out at sea and wishes to navigate using the North Star. At a particular moment, the sailor observes the North Star at an angle of elevation of 40 degrees from his boat. The boat is floating at a level height of 3 meters above the waterline. The sailor uses trigonometric principles to find out how far away he is from the point directly beneath the North Star. This helps in determining his route back to the shore.

MCQs:

1. What is the angle of elevation of the North Star?

   • A) 30 degrees
   • B) 40 degrees
   • C) 50 degrees
   • D) 60 degrees

2. What is the height of the sailor's boat above the waterline?

   • A) 1 meter
   • B) 2 meters
   • C) 3 meters
   • D) 4 meters

3. Which trigonometric ratio can help calculate the horizontal distance $d$ from the boat to the point directly below the North Star?

   • A) $\sin$
   • B) $\cos$
   • C) $\tan$
   • D) All of the above

4. What is the horizontal distance from the boat to the point directly beneath the North Star if the angle of elevation is 40 degrees?

   • A) $\frac{3}{\tan(40)}$ meters
   • B) $3\tan(40)$ meters
   • C) $3\sin(40)$ meters
   • D) $3\cos(40)$ meters