SECTION D — Case-Based/Source-Based Integrated Questions [4 Marks Each]
-
4 Marks
Q1. Case Study: The Lookout Tower
A forest fire department uses a high observation tower to monitor activity in the region. Two forest rangers are stationed at different points on flat ground relative to the tower. Ranger A stands at point $A$ and observes the top of the tower at an angle of elevation of $30^\circ$. Ranger B stands at point $B$, which is closer to the base of the tower on the same straight line, and observes the top of the tower at an angle of elevation of $60^\circ$. The distance between Ranger A and Ranger B is $100\text{ metres}$.
a) Represent the given situation with a clearly labeled rough sketch showing the heights, angles, and distances (1 Marks)
b) Express the height of the tower in terms of the distance from Ranger B to the base of the tower using appropriate trigonometric ratios (1 Marks)
c) Find the exact height of the observation tower (2 Marks)
-
4 Marks
Q2. Case Study: Designing a Playground Slide
An engineering firm is designing a children's slide for a public park. The safety guidelines specify that the ladder leading up to the platform should make an angle of $60^\circ$ with the ground, while the sliding chute itself should make an angle of $30^\circ$ with the ground to ensure a smooth deceleration. The top platform where the slide starts is supported at a height of $3\text{ metres}$ vertically above the ground.
a) Calculate the length of the ladder required to reach the platform (1 Marks)
b) Calculate the total ground distance covered from the base of the ladder to the end tip of the sliding chute. (1 Marks)
c) If the angle of the sliding chute is increased to $45^\circ$ while keeping the platform height fixed at $3\text{ metres}$, what would be the new length of the sliding chute? (2 Marks)
-
4 Marks
Q3. Case Study: The Hot Air Balloon
During a local cultural festival, a hot air balloon is tethered to the ground by a long rope and held vertically stationary. Two spectators, standing on opposite sides of the balloon along a straight line passing directly beneath it, observe the balloon. Spectator 1 observes the balloon at an angle of elevation of $45^\circ$, and Spectator 2 observes it at an angle of elevation of $60^\circ$. The distance between the two spectators is $50\left(1 + \frac{1}{\sqrt{3}}\right)\text{ metres}$.
a) Write down the trigonometric relations connecting the height of the balloon to the distance of each spectator from the point directly under the balloon (1 Marks)
b) Determine the height of the hot air balloon above the ground. (1 Marks)
c) Find the distance between Spectator 1 and the point directly under the balloon. (2 Marks)
-
4 Marks
Q4. Case Study: The Surveying Drone
A surveying drone is flying horizontally at a constant altitude of $50\sqrt{3}\text{ metres}$ above a straight highway. From the drone, the pilot looks down and measures the angles of depression of two milestone markers located on the highway on opposite sides of the drone. The angle of depression of milestone $P$ is $30^\circ$ and that of milestone $Q$ is $60^\circ$
a) Draw a neat diagram illustrating the drone, the highway, and the angles of depression (1 Marks)
b) Find the distance between milestone $P$ and the point on the highway directly beneath the drone (1 Marks)
c) Calculate the total distance between the two milestones $P$ and $Q$ (2 Marks)
-
4 Marks
Q5. Case Study: The Flagpole and the Sun
At a particular time of day, the shadow of a vertical flagpole of height $h$ metres is found to be $\sqrt{3}h$ metres long on horizontal ground. A stray kite gets stuck in the upper section of the flagpole at a point exactly half its total height.
a) What is the angle of elevation of the sun at this particular moment? (1 Marks)
b) Find the distance from the tip of the shadow on the ground to the point where the kite is stuck on the flagpole (1 Marks)
c) If the sun's altitude (angle of elevation) increases by $15^\circ$, what will be the new length of the shadow of the flagpole? (2 Marks)
-
4 Marks
Q6. Case Study: The Lighthouse and Ships
A lighthouse standing on a cliff by the sea is $100\text{ metres}$ high. From the top of the lighthouse, the angles of depression of two ships approaching directly towards each other in a straight line are observed to be $30^\circ$ and $45^\circ$.
a) Represent the angle of depression and angle of elevation equivalence using a geometric property (1 Marks)
b) Find the distance of the ship with the $45^\circ$ angle of depression from the base of the lighthouse (1 Marks)
c) Calculate the distance between the two ships. (2 Marks)
-
4 Marks
Q7. Case Study:The Billboard Structure
An advertising billboard is mounted on top of a building. A surveyor stands at a point on the ground $50\text{ metres}$ away from the base of the building. The angle of elevation of the bottom of the billboard is $30^\circ$, and the angle of elevation of the top of the billboard is $60^\circ$
a) Find the height of the building alone (from the ground to the bottom of the billboard). (1 Marks)
b) Find the total height from the ground to the top of the billboard (2 Marks)
c) Determine the vertical height (length) of the billboard itself (1 Marks)
-
4 Marks
Q8. Case Study: The Circular Race Track and Floodlight
A giant floodlight pole is erected at the exact center of a circular race track of radius $20\text{ metres}$. A runner is training on the track. At a certain instant, the angle of elevation of the top of the floodlight pole from the runner's position is $45^\circ$.
a) Find the height of the floodlight pole. (1 Marks)
b) If the runner sprints along the circumference of the track to a new position such that the angle of elevation of the top of the pole becomes $30^\circ$, find the straight-line distance the runner moved from the initial position to the new position along the ground (assuming a chord path or arc explanation) (3 Marks)
-
4 Marks
Q9. Case Study: The Tree Struck by Lightning
During a severe thunderstorm, a tall vertical tree is broken at a certain height above the ground such that its top touches the ground at a distance of $8\text{ metres}$ from its base, making an angle of $30^\circ$ with the ground.
a) State the trigonometric ratio best suited to find the height of the remaining part of the tree touching the ground (1 Marks)
b) Find the height of the part of the tree that broke off (the hypotenuse length) (2 Marks)
c) Calculate the original total height of the tree before it was broken. (1 Marks)
-
4 Marks
Q10. Case Study: The Ladder Against a Vertical Wall
A painter places a $10\text{ metre}$ long ladder against a vertical wall such that its foot stays on the horizontal floor. Initially, the ladder makes an angle of $60^\circ$ with the ground. Due to a slip on the floor, the bottom of the ladder slides away from the wall until the angle it makes with the ground becomes $30^\circ$.
a) Find the initial height up the wall that the top of the ladder reached (2 Marks)
b) Find the new height up the wall after the ladder slipped (1 Marks)
c) Calculate the total horizontal distance the foot of the ladder slid across the floor. (1 Marks)
-
4 Marks
Q11. Case Study: The River and Bridge Survey
A survey team wants to measure the width of a river. Standing on a bridge directly above the water level, they sight two reference pillars planted on opposite banks of the river straight across from each other. From a point on the bridge $20\text{ metres}$ horizontally to one side of the perpendicular drop, the angles of depression of pillars $X$ and $Y$ on opposite banks are measured as $45^\circ$ and $60^\circ$ respectively.
a) Draw a schematic diagram showing the bridge level, the reference point, and the pillars on the banks (1 Marks)
b) Find the depth/height of the bridge above the river bed level if required, or calculate the distance from the reference point to pillar $X$. (2 Marks)
c) Determine the total width of the river between pillar $X$ and pillar $Y$. (1 Marks)
-
4 Marks
Q12. Case Study: The Television Tower on a Hill
A television transmission tower is fixed on the peak of a small natural hill. From a point on the level ground $100\text{ metres}$ away from the base of the hill, the angle of elevation of the base of the tower is $30^\circ$, and the angle of elevation of the top of the tower is $60^\circ$.
a) Find the height of the hill (1 Marks)
b) Find the total height from the ground to the top of the television tower (2 Marks)
c) Calculate the exact height of the television tower alone. (1 Marks)
-
4 Marks
Q13. Case Study: The Skyscraper Window Cleaner
A window cleaning platform is suspended outside a skyscraper. A person on the street looks up at the cleaning platform at an angle of elevation of $45^\circ$. After the platform is hoisted vertically upwards by $20\text{ metres}$, the angle of elevation from the same spot on the street becomes $60^\circ$.
a) Set up equations relating the horizontal distance $x$ from the observer to the building and the initial height $h$ of the platform. (1 Marks)
b) Find the initial height of the platform above the ground. (2 Marks)
c) Determine the horizontal distance of the observer from the base of the skyscraper. (1 Marks)
-
4 Marks
Q14. Case Study: The Mountain Peak Observation
An observer on the deck of a cruise ship sights the peak of a coastal mountain. The angle of elevation of the mountain peak is $30^\circ$. After the ship sails $500\text{ metres}$ directly towards the base of the mountain along a straight line, the angle of elevation of the peak becomes $60^\circ$
a) Sketch the path of the ship and the two right-angled triangles formed (1 Marks)
b) Express the height of the mountain in terms of the final distance from the ship to the mountain base (1 Marks)
c) Calculate the height of the mountain peak above sea level. (2 Marks)
-
4 Marks
Q15. Case Study: The Kite Flying Competition
During a kite flying competition, two participants, Aarav and Meera, let out strings of length $100\text{ metres}$ and $200\text{ metres}$ respectively. Aarav's kite string makes an angle of $60^\circ$ with the horizontal ground, while Meera's kite string makes an angle of $30^\circ$ with the ground. Assuming both strings are taut and straight:
a) Calculate the vertical height of Aarav's kite above the ground. (2 Marks)
b) Calculate the vertical height of Meera's kite above the ground. (1 Marks)
c) Whose kite is flying higher, and by what vertical distance? (1 Marks)
-
4 Marks
Q16. Case Study: The Windmill Blade Rotation
A large industrial windmill has a tower of height $50\text{ metres}$ supporting a rotating turbine blade assembly at the top. An engineer standing on level ground $50\text{ metres}$ away from the base of the tower observes the tip of a blade when it is pointing directly vertically upwards. The angle of elevation is measured to be $60^\circ$.
a) Find the angle of elevation of the center hub of the turbine blades from the engineer's position (2 Marks)
b) Calculate the height of the blade tip above the ground when pointing straight up (1 Marks)
c) Determine the length of the windmill blade. (1 Marks)
-
4 Marks
Q17. Case Study: The Street Light and Pedestrian
A street light bulb is fixed at the top of a pole $6\text{ metres}$ high. A woman of height $1.5\text{ metres}$ walks away from the base of the streetlight pole along a straight path at a uniform speed of $1\text{ m/s}$
a) Using similar triangles or trigonometric relations of the angle made by the ray of light passing over her head, find the length of her shadow after walking $3\text{ seconds}$ (1 Marks)
b) Find the rate at which the length of her shadow increases (2 Marks)
c) What is the angle made by the light beam from the top of the pole to the tip of her shadow with the vertical pole when she is $3\text{ metres}$ away from the pole? (1 Marks)
-
4 Marks
Q18. Case Study: The Fortress Wall and Moat
A historical fortress is protected by a wide water moat. A scout on the outer bank of the moat observes the top of the fortress wall at an angle of elevation of $45^\circ$. Moving back by $10\text{ metres}$ away from the moat, the angle of elevation changes to $30^\circ$.
a) Express the height of the fortress wall in terms of the width of the moat (1 Marks)
b) Find the width of the moat. (2 Marks)
c) Find the height of the fortress wall (1 Marks)
-
4 Marks
Q19. Case Study: The Communication Satellite Dish Alignment
A large parabolic satellite dish is mounted on a concrete pedestal. To calibrate the receiver arm, technicians measure angles relative to the horizontal base plane. At a calibration test point $12\text{ metres}$ from the base of the pedestal, the angle of elevation to the edge of the dish rim is $30^\circ$, and to the focal feed horn it is $60^\circ$.
a) Calculate the height of the rim of the dish above the base level. (1 Marks)
b) Calculate the height of the focal feed horn above the base level. (2 Marks)
c) Find the vertical distance between the dish rim and the focal feed horn. (1 Marks)
-
4 Marks
Q20. Case Study: The Escort Boat and Submarine Periscope
An oceanographic research vessel uses a retractable periscope-like sensor. When extended, the sensor tip is $15\text{ metres}$ above the water surface. From the sensor tip, the angle of depression of a small marker buoy floating on the water is $30^\circ$.
a) Draw the right-angled triangle representing the sensor height, line of sight, and surface distance to the buoy (1 Marks)
b) Calculate the straight-line distance from the sensor tip to the floating 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.
Copyright © 2026 Deepa Maths Academy | Global Examination & Academic Portal. All Rights Reserved.