The given figure shows a hollow conical utensil and a hollow hemispherical utensil. The radius of each utensil is the same and when they are polished with the same kind of material, the cost of polishing them is also same.
What is the ratio of the depth of the hemispherical utensil and that of the conical utensil?
a) 2:1
b) √3:1
c) 1:√3
d) 1:2
Answers
Answer:
1:root 3
Step-by-step explanation:
by applying formula
Given :
- Radius of conical utensil = radius of hemispherical utensil
- Both utensil are made of same kind of material
- Cost of polishing conical utensil = cost of hemispherical utensil
To find :
Ratio of depth of hemispherical utensil to depth conical utensil
Formula used :
- Curved surface area of cone = πrl
- Curved surface area of hemisphere = 2πr²
Solution :
We know , cost of polishing both are same.
As they are made of same material , therefore curved surface area of both utensil will be same.
_______________________________________________
Let -
- Radius of conical utensil = r = Radius of hemispherical utensil
- Depth of conical utensil = d
- Slant depth of conical utensil = l
- Depth of hemispherical utensil = D
_______________________________________________
CSA of conical utensil = πrl
CSA of hemispherical utensil = 2πr²
As CSA of both are same
➝ πrl = 2πr²
➝ l = 2r......... equation 1
_______________________________________________
Now we know that , for cone
(slant depth)² = (depth)² + (radius)²
➝ l² = d² + r²
Put l = 2r from equation 1
➝ (2r)² = d² + r²
➝ 4r² = d² + r²
➝ d² = 4r² - r²
➝ d² = 3r²
➝ d = √(3r²)
➝ d = ± r√3
{ as , d denote side it cannot be negative }
Therefore , d = r√3
_______________________________________________
Depth of hemispherical utensil = radius of hemispherical utensil
➝ D = r
_______________________________________________
Ratio of the depth of the hemispherical utensil and that of the conical utensil
➝ D : d = r : r√3
➝ D : d = 1 : √3
______________________________________________
ANSWER :
Option c) 1 : √3