Video Transcript
Assuming that the value of 𝜋 is 22
over seven, find the radius of a sphere given its volume is 179.6 recurring cubic
centimeters.
We begin by recalling the formula
for the volume of a sphere. It is equal to four-thirds 𝜋𝑟
cubed. In this question, we are told that
the volume is equal to 179.6 recurring cubic centimeters. We know that 0.6 recurring is equal
to the fraction two-thirds. This means that the volume of the
sphere can be rewritten as 179 and two-thirds cubic centimeters. We can convert this mixed number
into an improper or top-heavy fraction. We do this by multiplying the whole
number by the denominator and then adding the numerator. 179 multiplied by three plus two is
equal to 539. The volume of the sphere is
therefore equal to 539 over three cubic centimeters. Substituting this value into our
formula together with the value of 𝜋 of 22 over seven, we have 539 over three is
equal to four-thirds multiplied by 22 over seven multiplied by 𝑟 cubed. Simplifying the right-hand side
gives us 88 over 21𝑟 cubed.
Next, we can multiply through by
three such that 539 is equal to 88 over seven 𝑟 cubed. Dividing through by 88 over seven
and then dividing the numerator and denominator by 11 gives us 𝑟 cubed is equal to
343 over eight. We can then take the cube root of
both sides of this equation. Noting that 343 and eight are both
perfect cubes, as two cubed is equal to eight and seven cubed is equal to 343, we
have 𝑟 is equal to the cube root of seven cubed over two cubed. Recalling one of the properties of
cube roots that if 𝑎 and 𝑏 are integers and 𝑏 is nonzero, the cube root of 𝑎
cubed over 𝑏 cubed is equal to 𝑎 over 𝑏, this means that 𝑟 is equal to seven
over two, which in decimal form is 3.5. If the volume of a sphere is 179.6
recurring cubic centimeters, then its radius is 3.5 centimeters.
