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IGCSE Class 10  >  Class 10 Notes  >  Mathematics for GCSE/  >  Case Based Questions: Surface Area and Volumes

Case Based Questions: Surface Area and Volumes

Case Study - 1

On a Sunday, your Parents took you to a fair. You could see lot of toys displayed, and you wanted them to buy a RUBIK's cube and strawberry ice-cream for you.
Observe the figures and answer the questions:
Case Study - 1Q1: The total surface area of cone with hemispherical ice cream is
(a) 858 cm2
(b) 885 cm2
(c) 588 cm2
(d) 855 cm2
Ans: (a)
Explanation: The total surface area of a cone with hemispherical ice cream is given by the formula: TSA = CSA of the cone + CSA of the hemisphere. The Curved Surface Area (CSA) of a cone is given by the formula: πrl, where r is the radius and l is the slant height. The CSA of a hemisphere is 2πr2. Therefore, the TSA is πr(l+2r). Assuming the radius to be 7cm, and the slant height to be 25cm (as given in Q4), the TSA = π*7*(25 + 2*7) = 22/7*7*(25 + 14) = 858cm2. 

Q2: What is the curved surface area of hemisphere (ice cream) if the base radius is 7cm?
(a) 309 cm2
(b) 308 cm2
(c) 803 cm2
(d) 903 cm2
Ans: (b)
Explanation: The curved surface area of a hemisphere is given by the formula: CSA = 2πr2. Here, the radius is given as 7cm. Therefore, the CSA = 2*22/7*7*7 = 308cm2. 

Q3: The length of the diagonal if each edge measures 6cm is
(a) 3√ 3
(b) 3√6
(c) √12
(d) 6√ 3
Ans: (d)
Explanation: The length of the diagonal of a cube is given by the formula: √3*a, where a is the length of an edge. Given that each edge measures 6cm, the diagonal = √3*6 = 6√3. 

Q4: Slant height of a cone if the radius is 7cm and the height is 24 cm
(a) 26cm
(b) 25 cm
(c) 52 cm
(d) 62cm
Ans: (b)
Explanation: The slant height of a cone can be found using the Pythagorean theorem: l = √(r2 + h2), where r is the radius and h is the height. Given that the radius is 7cm and the height is 24cm, the slant height = √(72 + 242) = 25cm.

Q5: Volume of the solid figure if the length of the edge is 7cm is
(a) 256 cm3
(b) 196 cm3
(c) 343 cm3
(d) 434 cm3
Ans: (c)
Explanation: The volume of a cube is given by the formula: V = a3, where a is the length of an edge. Given that each edge is 7cm, the volume = 73 = 343cm3. 


Case Study - 2

Adventure camps are the perfect place for the children to practice decision making for themselves without parents and teachers guiding their every move. Some students of a school reached for adventure at Sakleshpur. At the camp, the waiters served some students with a welcome drink in a cylindrical glass and some students in a hemispherical cup whose dimensions are shown below. After that they went for a jungle trek. The jungle trek was enjoyable but tiring. As dusk fell, it was time to take shelter. Each group of four students was given a canvas of area 551m2. Each group had to make a conical tent to accommodate all the four students. Assuming that all the stitching and wasting incurred while cutting, would amount to 1m2, the students put the tents. The radius of the tent is 7m.
Case Study - 2Case Study - 2

Q1: How much space on the ground is occupied by each student in the conical tent
(a) 54m2
(b) 38.5m2
(c) 86 m2
(d) 24 m2
Ans: (b)
Explanation: The area of the ground occupied by the conical tent is calculated using the formula for the area of a circle, πr².
Given the radius of the tent (r) is 7m, the area of the ground occupied by the tent is π * 7² = 154m².
Since each tent accommodates 4 students, the space on the ground occupied by each student is 154m² / 4 = 38.5m². Hence, the correct answer is option (b) 38.5m².

Q2: Which container had more juice and by how much?
(a) Hemispherical cup, 195 cm3
(b) Cylindrical glass, 207 cm3
(c) Hemispherical cup, 280.85 cm3
d) Cylindrical glass, 314.42 cm3
Ans: (d)
Explanation: The volume of the cylindrical glass is generally calculated by the formula πr²h and the volume of the hemispherical cup is calculated by the formula 2/3*πr³. By comparing these two volumes, we can determine which container had more juice. 

Q3: The volume of cylindrical cup is
(a) 295.75 cm3
(b) 7415.5 cm3
(c) 384.88 cm3
(d) 404.25 cm3
Ans: (d)
Explanation: The volume of the cylindrical cup is calculated by the formula πr²h. 

Q4: The volume of hemispherical cup is
(a) 179.67 cm3
(b) 89.83 cm3
(c) 172.25 cm3
(d) 210.60 cm3
Ans: (b)
Explanation: The volume of the hemispherical cup is calculated by the formula 2/3*πr³. 

Q5: The height of the conical tent prepared to accommodate four students is
(a) 18m
(b) 10m
(c) 24m
(d) 14m
Ans: (b)
Explanation: The height of the conical tent can be calculated if we have the volume and the radius of the base of the tent, using the formula for the volume of a cone: V=1/3*πr²h. By rearranging the formula, we can solve for h: h=3V/πr².

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FAQs on Case Based Questions: Surface Area and Volumes

1. How do I find the total surface area of a composite solid made from a cube and pyramid?
Ans. Total surface area of composite solids equals the sum of exposed surfaces of each shape, excluding areas where shapes connect. Identify which faces remain visible after joining, calculate their individual areas using formulas for cubes and pyramids, then add them together. This approach prevents counting hidden surfaces twice and is crucial for case-based problems involving real-world objects like roofs or containers.
2. What's the difference between lateral surface area and total surface area when solving volume and surface area case questions?
Ans. Lateral surface area covers only the curved or slanted sides of a solid, excluding top and bottom bases, whilst total surface area includes all outer surfaces. For cylinders and cones, lateral surface area = 2πrh and πrl respectively; total surface area adds the circular base(s). Understanding this distinction prevents calculation errors in CBSE case-based questions about paint coverage or material requirements.
3. How do I calculate the volume of irregular shapes in real-world case scenarios?
Ans. Break irregular shapes into simpler solids (cubes, cylinders, spheres, cones) whose volumes are known, calculate each component separately using standard formulas, then add or subtract volumes depending on whether shapes combine or overlap. This decomposition method works for water tanks, storage boxes, and architectural designs commonly featured in Class 10 surface area and volume case studies.
4. Why do my surface area answers differ when I use different formulas for the same shape?
Ans. Different formulas measure different parts of a shape-curved surface area versus total surface area including bases. Ensure you're using the correct formula for what the question asks. For instance, a cylinder's curved surface area (2πrh) differs from total surface area (2πrh + 2πr²). Always read case-based questions carefully to identify whether bases or only lateral surfaces count.
5. How can I quickly check if my volume and surface area calculations are reasonable for case-based questions?
Ans. Use estimation: compare your answer to simpler, similar shapes with known dimensions. Check dimensional consistency-volume should be in cubic units, surface area in square units. Verify that larger dimensions produce proportionally larger values. Reference mind maps and flashcards on EduRev to practise estimation techniques and double-check calculations before submitting answers in exam conditions.
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