raymond wants to make a box that has a volume of 360 cubic inches. He wants the height to be 10 inches and the other two dimensions to be whole numbers of inches. How many different-size boxes can he make?
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Volume = height * width * depth
You know the height is 10 inches, so the equation is 10 * x * x = 360.
Divide both sides by 10 to get x * x = 36.
The combinations to get a product of 36 are the dimensions.
To find out how many different-size boxes Raymond can make, we need to consider the factors of the volume of the box.
The volume of a rectangular box is calculated by multiplying its length, width, and height. In this case, the height is fixed at 10 inches. Let's denote the length as L and the width as W.
So, the volume equation becomes: V = L * W * 10
Given that the volume of the box is 360 cubic inches, we have 360 = L * W * 10.
To find the factors of 360, we can start by listing its prime factorization: 360 = 2^3 * 3^2 * 5^1.
Next, we can count the number of ways we can combine these prime factors to get two numbers, L and W, such that L * W * 10 equals 360.
The factors of 360 are:
1, 360 (1*360*10 = 3600)
2, 180 (2*180*10 = 3600)
3, 120 (3*120*10 = 3600)
4, 90 (4*90*10 = 3600)
5, 72 (5*72*10 = 3600)
6, 60 (6*60*10 = 3600)
8, 45 (8*45*10 = 3600)
9, 40 (9*40*10 = 3600)
10, 36 (10*36*10 = 3600)
12, 30 (12*30*10 = 3600)
15, 24 (15*24*10 = 3600)
18, 20 (18*20*10 = 3600)
There are 12 different combinations of length and width (L and W) that satisfy the given conditions, so Raymond can make 12 different-size boxes.
Note: The prime factorization method allows us to find all the factors of a given number easily. Another way to find the factors of 360 is through systematic trial division by starting with 1 and incrementing until the square root of 360. However, prime factorization is generally more efficient and reliable for finding factors.