Comparing the Volume of Two Spheres
The radius of sphere A is 2 cm, and the radius of sphere B is 4 cm. how many times larger is the volume of sphere B compared to the volume of sphere A?
The following is the formula for finding the volume of a sphere:
V = 4/3 πr
Now, the normal assumption is that we must use the formula to calculate the volume of both spheres in order to compare. However, observe that in comparing the volume of two different spheres, the only difference between them is the radius, thus we can compare the two spheres directly in this way:
(radius of sphere B)^3 / (radius of sphere A)^3
= 4^3 / 2^3
Therefore, the volume of sphere B is 8 times larger
than the volume of sphere A.
Hang on, you might say, why didn't we use the whole formula in comparing the volume of the two spheres? That's because of the properties of operations:
When we want to compare two terms, we express the comparison as fractions, thus to compare A with B, we can express the comparison as A/B, e.g., 8 compared with 4, 8/4 = 2, thus 8 is two times bigger than 4, or, 4 compared with 8, 4/8 = 1/2, thus 4 is half of 8.
Thus, comparing the volume of sphere B to the volume of sphere A can be expressed in this way:B
= volume of sphere B, and A
= volume of sphere A.
Next, there's also fraction equivalence, where a fraction a
is equivalent to a fraction (n
). For example, 2/3 = (3x2)/(3x3). The expression 4/3 πr
^3 can be separated as (4/3 π)(r
^3) Therefore, (4/3 πr1
^3) = r1
^3 / r2
Why didn't we remove ^3 then? Well, because ^3 means we are multiplying the number by itself twice, thus r
^3 = r
, that is why we cannot take out ^3 from the equation when comparing the volume of the two spheres.
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