Surface Areas and Volumes
Volume of sphere = 4/3*pie*r3
let the radius of the first sphere be 'R'
let the radius of the second sphere be 'r'As per the question,(4/3*pie*R3) / (4/3*pie*r3) = 64 / 27= R3/ r3= 64 / 27= R / r = 4 / 3 - - - - - - - (1)Now, CSA of a sphere = 4*pie*r2Ratio of CSA's of the two spheres is:(4*pie*R2) / (4*pie*r2)R2/ r2= (R / r)2Using (1)(R / r)2= (4/3)2=16 / 9Therefore, the ration of the CSA's of the two spheres is 16:9.Hope This Helps!!
arghh! That thing is so messed up .-. Lemme try again!
Volume of sphere = 4/3*pie*r3
let the radius of the first sphere be 'R'
let the radius of the second sphere be 'r'As per the question,
As perthe question
(4/3*pie*R3) / (4/3*pie*r3) = 64 / 27
= R3/ r3= 64 / 27= R / r = 4 / 3 - - - - - - - (1)
Now, CSA of a sphere = 4*pie*r2
Ratio of CSA's of the two spheres is:
(4*pie*R2) / (4*pie*r2)
R2/ r2= (R / r)2
Using (1)(R / r)2= (4/3)2=16 / 9
Therefore, the ratio of the CSA's of the two spheres is 16:9.
HopeThis Helps!!
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10 Questions 10 Marks 8 Mins
Given:
The ratio of the volume of the two spheres = 64 : 27
Formula used:
The volume of the sphere = (4/3) × π × R3
The surface area of the sphere = 4 × π × R2 Where R = The radius of the sphere
Calculation:
Let us assume the ratio of the surface area of the sphere be X : Y and the radius of the spheres be R1 and R2 respectively
⇒ The volume of the first sphere = [(4/3) × π × R13] ----(1)
⇒ The volume of the second cylinder = [(4/3) × π × R23] ----(2)
⇒ According to the question equation (1) ÷ (2) = 64 : 27
⇒ (R1/R2)3 = 64/27
⇒ R1/R2 = ∛(64/27)
⇒ R1/R2 = 4/3
⇒ Let us assume the radius of the first sphere = 4x and the second sphere = 3x
⇒ The surface area of the first sphere = 4 × π × (4x)2 = 64πx2 ----(3)
⇒ The surface area of the second sphere = 4 × π × (3x)2 = 36πx2 ----(4)
⇒ The ratio of the surface of the spheres = (64πx2)/(36πx2)
⇒ The ratio of the surface area of the spheres = 16/9
⇒ The ratio of their surface area X : Y = 16 : 9
∴ The required result will be 16 : 9.
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Text Solution
Solution : Let the radius of two spheres be `r_(1)` and `r_(2)` <br> Given, the ratio of the volume of two spheres = 64: 27 <br> `(V_(1))/(V_(2)) =(64)/(27) rArr ((4)/(3)pir_(1)^(3))/((4)/(3)pir_(2)^(3)) = (64)/(27)` <br> `rArr" "((r_(1))/(r_(2)))^(3) = ((4)/(3))^(3) " "[because "volume of sphere" =(4)/(3) pir^(3)]` <br> `rArr " "(r_(1))/(r_(2)) =(4)/(3)` <br> Let the surface areas of the two spheres `S_(1)` and `S_(2)` <br> `therefore" "(S_(1))/(S_(2)) = (4pir_(1)^(2))/( 4pir_(2)^(2)) = ((r_(1))/(r_(2)))^(2) rArr S_(1),S_(2) = ((4)/(3))^(2) = (16)/(9)` <br> `rArr" "S_(1),S_(2) = 16:9` <br> Hence, the ratio of the their surface areas is 16: 9.