Video Transcript
The side length, 𝑙, of a cube is
given by the formula 𝑙 is equal to the cube root of 𝑉, where 𝑉 is the volume of
the cube. What is the side length of a cube
whose volume is 5,832 cubic centimeters?
In this question, we are given the
formula for the side length of a cube using its volume. And we want to use this formula to
find the length of a side of a cube with a given volume. To find the length, we need to
start by substituting the volume of the cube into the formula. We get that 𝑙 is equal to the cube
root of 5,832.
To evaluate the cube root, we can
start by recalling that the cube root operation is the reverse operation to
cubing. In general, for any real number 𝑎,
the cube root of 𝑎 cubed is equal to 𝑎.
To use this result to evaluate the
cube root of the volume, we need to rewrite the volume as a cube. We can do this by factoring the
volume into primes. First, we note that the last digit
of the volume is two. So, we know that the number is
even. Hence, two is a factor of the
volume. We can divide the volume by two to
obtain 2,916. We can apply this same process two
more times to see that the volume has three factors of two and another factor of
729.
If we continue this process, we can
notice that the sum of the digits of 729 is a multiple of nine. So, it is divisible by nine. In fact, it is equal to nine
cubed. So, we have shown that 5,832 is
equal to two cubed times nine cubed.
Either by expanding the product or
by using the laws of exponents, we can then rewrite this as 18 cubed. Hence, the length of the cube is
equal to the cube root of 18 cubed. We can then evaluate this to get 18
centimeters.
It is worth noting that we can
verify our answer by recalling that the volume, 𝑉, of a cube with sides of length
𝑙 is given by the formula 𝑉 is equal to 𝑙 cubed. If we substitute a length of 18
centimeters into the formula, we see that the volume of the cube is 18 cubed cubic
centimeters, which we can calculate is 5,832 cubic centimeters. This is the same as the volume
given in the question, confirming that its sides must have length 18
centimeters.
