Question 34 (Choice 2) - CBSE Class 10 Sample Paper for 2023 Boards - Maths Standard - Solutions of Sample Papers for Class 10 Boards

Last updated at April 16, 2024 by Teachoo

There are two identical solid cubical boxes of side 7cm. From the top face of the first cube a hemisphere of diameter equal to the side of the cube is scooped out. This hemisphere is inverted and placed on the top of the second cube’s surface to form a dome. Find

(i) the ratio of the total surface area of the two new solids formed

(ii) volume of each new solid formed.

Transcript

Question 34 (Choice 2) There are two identical solid cubical boxes of side 7cm. From the top face of the first cube a hemisphere of diameter equal to the side of the cube is scooped out. This hemisphere is inverted and placed on the top of the second cube’s surface to form a dome. Find (i) the ratio of the total surface area of the two new solids formed (ii) volume of each new solid formed.
Question 34 (Choice 2) – Part (i) Find (i) the ratio of the total surface area of the two new solids formed
Solid I
Here,
Side of Cube = a = 7 cm
Radius of Hemisphere = 7/2 cm
Total Surface Area of Solid I
= Total Area of cube + Curved surface area of hemisphere − Base area of hemisphere
= 6a2 + 2𝜋r2 − 𝜋r2
= 6a2 + 𝜋r2
= 6 × 72 + 22/7×(7/2)^2
= 6 × 49 + 22/7×7/2 ×7/2
= 294 + 11 × 7/2
= 294 + 38.5
= 332.5 cm2
Solid II
Total Surface Area of Solid II
= Total Area of cube + Curved surface area of hemisphere − Base area of hemisphere
= 6a2 + 2𝜋r2 − 𝜋r2
= 6a2 + 𝜋r2
= 6 × 72 + 22/7×(7/2)^2
= 6 × 49 + 22/7×7/2 ×7/2
= 294 + 11 × 7/2
= 294 + 38.5
= 332.5 cm2
Since Total Surface Area of both the Solids are 332.5 cm2
Therefore,
Ratio of their Surface Area = 332.5 : 332.5
= 1 : 1Question 34 (Choice 2) – Part (ii) Find (ii) volume of each new solid formed.Now,
Volume of Solid I = Volume of Cube − Volume of hemisphere
And
Volume of Solid II = Volume of Cube + Volume of hemisphere
Volume of Cube
Side of Cube = a = 7 cm
Volume of cube = a3
= 73
= 7 × 7 × 7
= 343 cm3
Volume of Hemisphere
Radius of Hemisphere = 7/2 cm
Volume of Hemisphere = 2/3 𝜋𝑟^3
= 2/3 ×22/7 × (7/2)^3
= 2/3 ×22/7 ×7/2 ×7/2 ×
7/2 7/2= 2/3 ×22/7 ×7/2 ×7/2 × 7/2
= 539/6
= 89.83 cm3
Thus,
Volume of Solid I = Volume of Cube − Volume of hemisphere
= 343 − 89.83
= 253.17 cm3
And
Volume of Solid II = Volume of Cube + Volume of hemisphere
= 343 + 89.83
= 432.83 cm3

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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