if the ratio of the diameters of two spheres is A to B then what is the ratio of their volumes
volume is proportional to radius cubed, so
Va/Vb=(A/B)^3
To find the ratio of the volumes of two spheres, knowing the ratio of their diameters, we'll use the formula for the volume of a sphere, which is V = (4/3)πr^3, where V represents the volume and r is the radius.
Let's assume the diameters of the spheres are D1 and D2, and the corresponding radii are r1 and r2. The ratio of their diameters can be expressed as A:B, so D1/D2 = A/B.
To find the relationship between the radii, we need to divide the diameter by 2. Therefore, r1 = D1/2 and r2 = D2/2.
To find the ratio of the volumes, we substitute the values into the volume formula:
V1 = (4/3)πr1^3
V2 = (4/3)πr2^3
Substituting for r1 and r2:
V1 = (4/3)π(D1/2)^3
V2 = (4/3)π(D2/2)^3
Simplifying:
V1 = (4/3)π(D1^3/8)
V2 = (4/3)π(D2^3/8)
Now, we can find the ratio of the volumes by dividing V1 by V2:
V1/V2 = [(4/3)π(D1^3/8)] / [(4/3)π(D2^3/8)]
Canceling out the common factors:
V1/V2 = (D1^3/8) / (D2^3/8)
Simplifying further:
V1/V2 = D1^3 / D2^3
Hence, the ratio of the volumes of the two spheres is (D1^3 / D2^3) or (A^3 / B^3).