SECTION D — Case-Based/Source-Based Integrated Questions [4 Marks Each]
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4 Marks
Q1. Case Study: The Airport Air Traffic Control Tower
The Air Traffic Control (ATC) tower at an international airport is designed to give controllers a clear view of the runways. From the observation cabin at the top of the tower, $40\text{ metres}$ above ground level, an air traffic controller observes an incoming aircraft on the runway at an angle of depression of $30^\circ$ and a service vehicle parked near the taxiway at an angle of depression of $60^\circ$ in the opposite direction along a straight line.
a) Represent the given situation with a neat, labeled geometric diagram (1 Marks)
b) Calculate the horizontal distance of the service vehicle from the base of the ATC tower (2 Marks)
c) Find the straight-line ground distance between the aircraft and the service vehicle, assuming they lie on a straight line passing through the base of the tower (1 Marks)
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4 Marks
Q2. Case Study: The Telecommunication Tower and Building
A telecommunication company installs a transmission tower on the roof of a commercial building. A quality inspector standing on the flat street $50\text{ metres}$ away from the base of the building observes the angle of elevation to the bottom of the transmission tower to be $45^\circ$, and the angle of elevation to the top of the transmission tower to be $60^\circ$
a) Find the height of the commercial building from the ground to the base of the tower (2 Marks)
b) Determine the total height of the structure from the ground to the top of the transmission tower (1 Marks)
c) Find the exact vertical height of the transmission tower alone. (1 Marks)
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4 Marks
Q3. Case Study: The Highway Overpass and Moving Vehicle
An engineering student stands on a highway overpass bridge and observes cars moving below. From a point on the bridge directly above a reference pillar, the angle of depression of a stationary delivery truck parked on the service lane is $45^\circ$. A moving car traveling at a uniform speed is observed at an angle of depression of $30^\circ$, which after $10\text{ seconds}$ moves to a position where its angle of depression becomes $60^\circ$. The height of the overpass above the road level is $15\sqrt{3}\text{ metres}$
a) Find the horizontal distance of the stationary delivery truck from the point directly below the observer (1 Marks)
b) Calculate the distance traveled by the car during the $10\text{-second}$ interval (1 Marks)
c) Determine the speed of the moving car in $\text{m/s}$. (2 Marks)
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4 Marks
Q4. Case Study: The River Bridge and Surveying Pillars
A survey team measures the width of a river using trigonometry. A suspension bridge crosses the river at an angle of $45^\circ$ to the river bank. The length of the bridge from one bank to the other is $120\text{ metres}$. Two reference markers $P$ and $Q$ are planted on opposite banks directly across from each other.
a) Sketch the layout showing the river banks, the bridge path, and the right-angled projection (1 Marks)
b) Calculate the actual perpendicular breadth of the river. (1 Marks)
c) If an observer stands on the bridge exactly at its midpoint, find its straight-line height above the water level assuming the bridge profile rises uniformly to a maximum height of $15\text{ metres}$ at the center (2 Marks)
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4 Marks
Q5. Case Study: The Hot Air Balloon Festival
During a local science festival, a hot air balloon is tethered securely using a long steel cable. Two spectators standing on level ground on opposite sides of the balloon along a straight line passing directly underneath it measure its elevation. Spectator A measures the angle of elevation as $30^\circ$ and Spectator B measures it as $60^\circ$. The distance between Spectator A and Spectator B is $80\text{ metres}$.
a) Write down the trigonometric relationship linking the height $h$ of the balloon to the distance of Spectator A from the point under the balloon (1 Marks)
b) Calculate the exact vertical height of the hot air balloon above the ground (2 Marks)
c) Find the length of the tether cable if it runs straight from the balloon to Spectator B (1 Marks)
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4 Marks
Q6. Case Study: The Broken Palm Tree
During a cyclone, a tall palm tree standing on level ground is fractured at a certain height above the ground. The upper portion bends downwards without breaking completely off, so that its topmost crown touches the horizontal ground at a distance of $6\sqrt{3}\text{ metres}$ from the base of the tree, making an angle of $60^\circ$ with the ground.
a) Find the height of the remaining vertical portion of the tree attached to the ground. (1 Marks)
b) Calculate the length of the broken part of the tree that touches the ground (2 Marks)
c) Determine the original total height of the palm tree before the storm (1 Marks)
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4 Marks
Q7. Case Study:The Mountain Peak and Cruise Ship
Passengers on a luxury cruise ship observe a prominent peak of a coastal mountain. At 10:00 AM, the angle of elevation of the mountain peak from the ship deck is $30^\circ$. The ship sails directly towards the shore at a uniform speed for $20\text{ minutes}$, after which the angle of elevation of the peak is measured to be $60^\circ$. The mountain peak is known to be $1500\text{ metres}$ above sea level.
a) Find the distance of the ship from the base of the mountain at the second observation point ($60^\circ$ angle) (2 Marks)
b) Find the distance of the ship from the base of the mountain at the initial observation point ($30^\circ$ angle). (1 Marks)
c) Calculate the speed at which the cruise ship was sailing in $\text{m/min}$ (1 Marks)
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4 Marks
Q8. Case Study: The Street Light and Pedestrian Shadow
A street light is mounted at the top of a pole $9\text{ metres}$ high. A woman of height $1.8\text{ metres}$ walks away from the base of the pole along a straight horizontal path at a speed of $1.2\text{ m/s}$.
a) Using similar triangles or trigonometric ratios, set up the relation for the length of her shadow when she is $4\text{ metres}$ away from the pole (2 Marks)
b) Find the length of her shadow exactly $5\text{ seconds}$ after she starts walking away from the pole (1 Marks)
c) Calculate the rate at which the length of her shadow increases (1 Marks)
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4 Marks
Q9. Case Study: The Multi-Storey Building and Flagstaff
A historical building is topped with a decorative flagstaff. An architect stands on level ground at a certain distance from the building. The angle of elevation of the base of the flagstaff is $30^\circ$, and the angle of elevation of the top of the flagstaff is $60^\circ$. The building itself is $15\text{ metres}$ high.
a) Calculate the horizontal distance of the architect from the base of the building (2 Marks)
b) Find the total height from the ground to the top of the flagstaff. (1 Marks)
c) Determine the exact length of the flagstaff alone (1 Marks)
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4 Marks
Q10. Case Study: The Jet Fighter Formation
During an air show display, a jet fighter flies at a constant horizontal altitude of $2400\sqrt{3}\text{ metres}$ above the ground. An observer at point $O$ on the ground sights the jet fighter at an angle of elevation of $60^\circ$. After a duration of $20\text{ seconds}$, the observer looks back and finds the angle of elevation has changed to $30^\circ$ as the jet flies away in the same straight line.
a) Find the horizontal distance of the jet from the observer at the initial instant (2 Marks)
b) Find the horizontal distance of the jet from the observer after $20\text{ seconds}$. (1 Marks)
c) Calculate the uniform speed of the jet fighter in $\text{km/h}$ (1 Marks)
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4 Marks
Q11. Case Study: The Lake, Cloud, and Reflection
A nature photographer standing on a cliff $20\text{ metres}$ above the calm surface of a mountain lake observes a cloud in the sky. The angle of elevation of the cloud is $30^\circ$, and the angle of depression of its mirror reflection in the lake water is $60^\circ$.
a) Let the height of the cloud above the lake surface be $H$. Express the height of the reflection below the lake surface. (1 Marks)
b) Set up the trigonometric equation connecting $H$, the observer height, and the angle of elevation (2 Marks)
c) Calculate the exact height of the cloud above the lake surface. (1 Marks)
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4 Marks
Q12. Case Study: The Hillside Slope and Transmission Post
A transmission post is erected on the sloping side of a hill that makes an angle of $30^\circ$ with the horizontal. A surveyor standing $50\text{ metres}$ down the slope from the base of the post measures the angle of elevation of the top of the post (relative to the slope) to be $30^\circ$.
a) Draw a clear diagram showing the hill slope, the base, and the line of sight (1 Marks)
b) Determine the height of the transmission post perpendicular to the hill slope (2 Marks)
c) Find the straight-line distance from the surveyor's point to the top of the transmission post. (1 Marks)
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4 Marks
Q13. Case Study: The Lighthouse and Two Approaching Boats
A lighthouse stands on a sheer sea cliff, towering $100\text{ metres}$ above sea level. From the top observation deck, the angles of depression of two boats approaching the cliff along a straight line directly towards each other are observed to be $30^\circ$ and $45^\circ$.
a) Calculate the distance of the boat with the $45^\circ$ angle of depression from the base of the cliff (1 Marks)
b) Calculate the distance of the boat with the $30^\circ$ angle of depression from the base of the cliff (2 Marks)
c) Find the distance separating the two boats at this instant (1 Marks)
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4 Marks
Q14. Case Study: The Stunt Kite and Competition Grounds
At an international kite-flying competition, two contestants, Rohit and Priya, fly their kites. Rohit's kite string is $100\text{ metres}$ long and makes an angle of $60^\circ$ with the ground. Priya's kite string is $200\text{ metres}$ long and makes an angle of $30^\circ$ with the ground. Both strings are taut.
a) Calculate the vertical height of Rohit's kite above the ground. (2 Marks)
b) Calculate the vertical height of Priya's kite above the ground. (1 Marks)
c) Determine whose kite is flying higher and find the vertical difference between them. (1 Marks)
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4 Marks
Q15. Case Study: The Fortress Moat and Defensive Wall
A historical fortress is surrounded by a water moat. A scout standing on the outer edge of the moat measures the angle of elevation of the top of the fortress wall as $45^\circ$. Moving back by a distance of $20\text{ metres}$ directly away from the moat, the angle of elevation of the top of the wall becomes $30^\circ$.
a) Express the height of the fortress wall in terms of the initial width of the moat (1 Marks)
b) Calculate the width of the moat (2 Marks)
c) Find the exact height of the fortress wall (1 Marks)
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4 Marks
Q16. Case Study: The Window Cleaner and Skyscraper
A window cleaning platform is suspended outside a high-rise office building. A pedestrian on the sidewalk looks up at the platform at an angle of elevation of $45^\circ$. After the winch hoists the platform vertically upwards by $15\text{ metres}$, the angle of elevation from the exact same spot on the sidewalk changes to $60^\circ$.
a) Formulate algebraic equations relating the horizontal distance from the pedestrian to the building and the initial height of the platform (2 Marks)
b) Calculate the initial height of the cleaning platform above the ground (1 Marks)
c) Find the horizontal distance of the pedestrian from the base of the skyscraper. (1 Marks)
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4 Marks
Q17. Case Study: The Statue on a Pedestal
A bronze statue of height $1.6\text{ metres}$ is installed on top of a concrete pedestal. From a point on the horizontal ground, the angle of elevation of the top of the statue is $60^\circ$, and from the same point, the angle of elevation of the top of the pedestal is $45^\circ$.
a) Draw a schematic diagram showing the ground point, pedestal, statue, and angles. (1 Marks)
b) Find the height of the concrete pedestal. (2 Marks)
c) Determine the distance of the point on the ground from the base of the pedestal. (1 Marks)
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4 Marks
Q18. Case Study: The Forest Fire Lookout Towers
Two forest fire lookout towers, Tower P and Tower Q, are situated on flat terrain $100\text{ metres}$ apart. A ranger standing at a point on the ground between the towers on the straight line joining their bases observes the top of Tower P at an angle of elevation of $30^\circ$ and the top of Tower Q at an angle of elevation of $60^\circ$. The height of Tower P is known to be $\frac{100}{\sqrt{3}}\text{ metres}$.
a) Find the distance of the ranger from the base of Tower P. (2 Marks)
b) Find the distance of the ranger from the base of Tower Q. (1 Marks)
c) Calculate the height of Tower Q. (1 Marks)
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4 Marks
Q19. Case Study: The Windmill Blade and Hub
An industrial wind turbine features a tower of height $60\text{ metres}$ supporting a large rotating rotor hub at the top. An engineer standing on level ground $60\text{ metres}$ away from the base of the tower measures the angle of elevation of the rotor hub. When one of the turbine blades points directly vertically upwards, its tip is observed at an angle of elevation of $60^\circ$.
a) Find the angle of elevation of the central rotor hub. (1 Marks)
b) Calculate the total height of the blade tip above the ground when pointing straight up (2 Marks)
c) Determine the exact length of the turbine blade (1 Marks)
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4 Marks
Q20. Case Study: The Submarine Periscope and Marker Buoy
An oceanographic research vessel deploys an electronic sensor periscope. When fully extended, the sensor head is $20\text{ metres}$ above the calm water surface. From the sensor head, the angle of depression of a small calibration marker buoy floating on the water is $30^\circ$.
a) Sketch the right-angled triangle representing the sensor height, line of sight, and water surface distance. (1 Marks)
b) Calculate the straight-line distance from the sensor head to the marker buoy (1 Marks)
c) Find the horizontal distance from the research vessel to the marker buoy (2 Marks)
Academic Repository Disclaimer & எச்சரிக்கை அறிவிப்பு :
English: These model solutions are structured to global academic standards for self-assessment and practice purposes. Students are advised to verify with official textbooks for board examinations.
தமிழ்: இந்த மாதிரி கணிதத் தீர்வுகள் மாணவர்களின் சுய கற்றல் மற்றும் பயிற்சித் திறனுக்காக உலகளாவிய கல்வித் தரத்தில் உருவாக்கப்பட்டுள்ளன. தேர்வுகளுக்குத் தயாராகும் போது உங்கள் அதிகாரப்பூர்வ பாடப்புத்தகத்துடன் சரிபார்க்கவும்.
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