A cubical block of side 7 cm is surmounted by a hemisphere. What is the greatest diameter the hemisphere can have? Find the surface area of the solid
Solution:
We can create the following figure of the solid as per the given information.
From the figure, it’s clear that the greatest diameter the hemisphere can have is equal to the cube’s edge. Otherwise, a part of the hemisphere’s rim (circumference of its circular base) will lie outside the top part of the cube.
As it’s clear from the top view of the figure that the top part of the cube is partially covered by the hemispherical region.
Total surface area of the solid = Surface area of the cubical part - Area of the base of the hemispherical part + CSA of the hemispherical part
We will find the total area of the solid by using the formulae;
CSA of the hemisphere = 2πr2
Area of the base of the hemisphere = πr2, where r is the radius of the hemisphere
Surface area of the cube = 6l2, where l is the length of the edge of the cube
Length of the edge of the cube, l = 7 cm
From the figure, it’s clear that the greatest diameter the hemisphere can have is equal to the cube’s edge
Diameter of the hemisphere, d = l = 7 cm
Radius of the hemisphere, r = d/2 = 7/2 cm
Total surface area of the solid = Surface area of the cubical part - Area of the base of the hemispherical part + CSA of the hemispherical part
= 6l2 - πr2 + 2πr2
= 6l2 + πr2
= 6 × (7 cm)2 + 22/7 × (7/2 cm)2
= 6 × 49 cm2 + 22/7 × 49/4 cm2
= 294 cm2 + 38.5 cm2
= 332.5 cm2
Thus, the greatest diameter the hemisphere can have is 7 cm and the surface area of the solid is 332.5 cm2.
☛ Check: NCERT Solutions for Class 10 Maths Chapter 13
Video Solution:
A cubical block of side 7 cm is surmounted by a hemisphere. What is the greatest diameter the hemisphere can have? Find the surface area of the solid.
NCERT Solutions Class 10 Maths Chapter 13 Exercise 13.1 Question 4
Summary:
The greatest diameter of a hemisphere which surmounts a cubical block of side 7 cm is 7 cm and the surface area of the solid is 332.5 cm2.
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