Video Transcript
Find the diameter of a sphere given
that its volume is 135.5 centimeters cubed, taking the approximation 𝜋 equals 22
over seven. Give your answer to the nearest two
decimal places.
Recall that the volume of a sphere,
𝑉, is given by four over three times 𝜋 times its radius cubed. The diameter of a sphere, 𝑑, is
equal to twice its radius, 𝑟. Therefore, the radius, 𝑟, is equal
to the diameter, 𝑑, over two. So we can express the volume of a
sphere as four over three times 𝜋 times 𝑑 over two all cubed. Taking the cube of the parentheses
gives us 𝑑 cubed over eight, and this expression simplifies to one over six times
𝜋 times 𝑑 cubed.
We can rearrange this equation for
the diameter, 𝑑, by multiplying both sides by six, dividing both sides by 𝜋, and
taking the cube root, giving 𝑑 equals the cube root of six 𝑉 over 𝜋. Taking the approximation 𝜋 equals
22 over seven, dividing by 22 over seven is equivalent to multiplying by seven over
22. So we have 𝑑 equals the cube root
of six 𝑉 times seven over 22. And this simplifies to the cube
root of 21𝑉 over 11.
Inserting the value for 𝑉, the
volume of the sphere, we obtain 𝑑 equals the cube root of 21 times 135.5 over
11. Performing this calculation gives
us the diameter of the sphere, 6.37 to two decimal places. And the unit is centimeters.
