The dimension of a rectangular prism are all positive integers. The volume of the prism is 2002 cubic cm. Find the minimum possible sum in cm of the length width and height.
Factorize 2002 to get
2002=2*7*11*13
The three dimensions of the prism must be either one of the above factors, or a product of two of the factors, such as 14*11*13, or
22*7*13, or
2*77*13, etc
Can you find the dimensions which give the minimum sum?
To solve this problem, we need to find the positive integer dimensions of the rectangular prism that have a volume of 2002 cubic cm, and then find the minimum possible sum of the length, width, and height.
Let's start by listing the factors of 2002: 1, 2, 7, 11, 13, 14, 22, 26, 77, 91, 143, 154, 286, 429, 1001, and 2002.
Now we need to find three factors whose product is 2002. We can start by trying the smallest factors and work our way up:
1 x 2 x 1001 = 2002, but 1 + 2 + 1001 = 1004, which is not the minimum sum.
1 x 7 x 286 = 2002, and 1 + 7 + 286 = 294, but this is still not the minimum sum.
1 x 11 x 182 = 2002, and 1 + 11 + 182 = 194.
1 x 13 x 154 = 2002, and 1 + 13 + 154 = 168.
1 x 14 x 143 = 2002, and 1 + 14 + 143 = 158.
2 x 7 x 143 = 2002, and 2 + 7 + 143 = 152, which is the minimum possible sum.
Therefore, the minimum possible sum in cm of the length, width, and height is 152 cm.