Class 10 Maths Chapter 12 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:
Q1: 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.

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.