Create a real-world problem involving a cube - Use a perfect cube as its volume - Show using cube roots to find one edge
1 answer:
PART A
A box of sugar has equal lengths of 6 inches. Calculate the volume of sugar that can fill the box.
The volume of the box can be calculated using the cubic volume formula given to be:
Therefore, the volume of the box of sugar is calculated to be:
PART B
An ice cube is said to contain a volume of 8 cubic inches of water. What will be the length of one side of the cube?
The length of the cube can be calculated using the formula:
Hence, we can solve to be:
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There are a total of 2 + 3 + 6 = 11 letters, which gives rise to 11! possible permutations.
Consider one such permutation:
BBHHHOOOOOO
I write one of the Bs in the bold for emphasis. This permutation is not distinguishable (but for the temporary boldface) from
BBHHHOOOOOO
which is to say they both count as the same permutation. To avoid counting this twice, you would divide the total number of permutations by the number of ways you can permute the identical character. In the case of Bs, this can be done in 2! = 2 ways.
Similarly, the 3 Hs can be rearranged in 3! = 6 ways, and the 6 Os can be rearranged in 6! = 720 ways.
Then the total number of distinguishable permutations is
11! / (2! • 3! • 6!) = 4620
As a formula: given a word of length with
different characters that respectively occur
times, the total number of distinct permutations is
where .
Try this with some simple examples:
• GREECE
This has length 6, with E occurring three times, and every other letter occurs once. The total number of distinct permutations of GREECE is
6! / (1! • 1! • 1! • 3!) = 120
• BANANA
This also has length 6, with B occurring once, N twice, and A three times, so the total number of distinct permutations of BANANA is
6! / (1! • 2! • 3!) = 60
• STATISTICALLY
This has length 13, with one each of C and Y; two each of S, A, I, L; and three Ts. Then the number of distinct permutations is
13! / (1! • 1! • 2! • 2! • 2! • 2! • 3!) = 64,864,800