Assertion (A) : If surface areas of the two spheres are in the ratio 16 : 9, then their volumes are in the ratio 64 : 27.
Reason (R) : If S1 and S2 are the surface areas of two spheres and V1 and V2 are their volumes respectively, then V2V1=(S2S1)3/2.
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Step-by-Step Solution
Step 1: Relate surface area and radius
The surface area of a sphere is given by the formula S=4πr2, where r is the radius of the sphere. If we have two spheres with radii r1 and r2, their surface areas S1 and S2 will be S1=4πr12 and S2=4πr22.
Step 2: Find the ratio of radii
Given the ratio of surface areas S2S1=916, we can find the ratio of their radii. By dividing the surface area formulas, we get S2S1=(r2r1)2. Taking the square root of both sides gives us the ratio of the radii.
Step 3: Relate volume and radius
The volume of a sphere is given by the formula V=34πr3. For two spheres with radii r1 and r2, their volumes V1 and V2 will be V1=34πr13 and V2=34πr23.
Step 4: Find the ratio of volumes
We can find the ratio of the volumes by dividing their formulas. This simplifies to V2V1=(r2r1)3.
Step 5: Substitute ratio of radii into ratio of volumes
Now we substitute the expression for the ratio of radii, r2r1=S2S1, into the ratio of volumes. This gives us V2V1=(S2S1)3/2. This confirms that Reason (R) is a correct statement.
Step 6: Evaluate the assertion
Given that the ratio of surface areas is 16:9, we can use the derived relationship to find the ratio of volumes. Substituting S2S1=916 into the formula, we get V2V1=(916)3/2. This evaluates to 2764, which means the Assertion (A) is also correct.