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The volumes of two spheres are in the ratio 125 : 64. The ratio of their surface areas is
- a)9 : 16
- b)16 : 9
- c)25 : 16
- d)16 : 25
Correct answer is option 'C'. Can you explain this answer?
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The volumes of two spheres are in the ratio 125 : 64. The ratio of the...
Let r1 and r2 be the radius of the two spheres respectively. Therefore, the ratio of their surface areas,
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The volumes of two spheres are in the ratio 125 : 64. The ratio of the...
To solve this problem, we need to use the formulas for the volume and surface area of a sphere.
Let's assume that the volumes of the two spheres are V1 and V2, and their surface areas are A1 and A2, respectively.
The formula for the volume of a sphere is given by V = (4/3)πr^3, where r is the radius of the sphere. Similarly, the formula for the surface area of a sphere is given by A = 4πr^2.
Given that the volumes are in the ratio 125:64, we can write:
V1/V2 = 125/64
Now, let's find the ratio of their radii. We can use the fact that the volume of a sphere is proportional to the cube of its radius.
(V1/V2)^(1/3) = (125/64)^(1/3)
(1/2) = (5/4)^(1/3)
To simplify this, we can cube both sides:
(1/2)^3 = (5/4)^(1/3)^3
1/8 = 5/4
From this, we can see that the ratio of the radii of the two spheres is 1:2.
Now, let's find the ratio of their surface areas. We can use the fact that the surface area of a sphere is proportional to the square of its radius.
(A1/A2) = (r1^2)/(r2^2)
(A1/A2) = (1^2)/(2^2)
(A1/A2) = 1/4
To simplify this ratio, we can multiply the numerator and denominator by 4:
(A1/A2) = (1/4) * 4/4
(A1/A2) = 4/16
So, the ratio of their surface areas is 4:16, which simplifies to 1:4.
Therefore, the correct answer is option C) 25:16.
Let's assume that the volumes of the two spheres are V1 and V2, and their surface areas are A1 and A2, respectively.
The formula for the volume of a sphere is given by V = (4/3)πr^3, where r is the radius of the sphere. Similarly, the formula for the surface area of a sphere is given by A = 4πr^2.
Given that the volumes are in the ratio 125:64, we can write:
V1/V2 = 125/64
Now, let's find the ratio of their radii. We can use the fact that the volume of a sphere is proportional to the cube of its radius.
(V1/V2)^(1/3) = (125/64)^(1/3)
(1/2) = (5/4)^(1/3)
To simplify this, we can cube both sides:
(1/2)^3 = (5/4)^(1/3)^3
1/8 = 5/4
From this, we can see that the ratio of the radii of the two spheres is 1:2.
Now, let's find the ratio of their surface areas. We can use the fact that the surface area of a sphere is proportional to the square of its radius.
(A1/A2) = (r1^2)/(r2^2)
(A1/A2) = (1^2)/(2^2)
(A1/A2) = 1/4
To simplify this ratio, we can multiply the numerator and denominator by 4:
(A1/A2) = (1/4) * 4/4
(A1/A2) = 4/16
So, the ratio of their surface areas is 4:16, which simplifies to 1:4.
Therefore, the correct answer is option C) 25:16.
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The volumes of two spheres are in the ratio 125 : 64. The ratio of their surface areas isa)9 : 16b)16 : 9c)25 : 16d)16 : 25Correct answer is option 'C'. Can you explain this answer?
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