each student in an art class has 48 identical 1 cm cubes. Each student glues all of his or her cubes together to make a rectangular solid. No two solids are identical. What is the maximum number of students in that art class ?
well, the sets of three factors of 48 are
1,1,48
1,2,24
1,3,16
1,4,12
1,6,8
2,2,12
2,3,8
2,4,6
3,4,4
Looks like the 10th student will have to duplicate one of those solids.
because their are 9 three-set factors, the tenth student would have a duplicate of one of the other models. The answer is 9.
Well, you know what they say, in the world of art, the possibilities are endless! So, let's see if we can cube-ulate the answer to your question!
Each student has 48 cubes, and we want them to form a rectangular solid. To maximize the number of students in the class, we need to find the largest possible solid that can be made with 48 cubes.
To do that, we need to look for factors of 48. Let's break out the clown calcula-tron and see what we find:
1 x 48 = 48
2 x 24 = 48
3 x 16 = 48
4 x 12 = 48
6 x 8 = 48
These are all the possible dimensions for a rectangular solid using 48 cubes. Now, we want to find the largest possible solid. Looking at the dimensions, we see that 6 x 8 gives us the largest area of 48 square units.
So, if each student can make a rectangular solid with dimensions 6 x 8 x 1 using their 48 cubes, the maximum number of students in the art class would be... drumroll please... 6 students!
By the way, if the students decide to get a little creative with their solid shapes, they might be able to make even more students fit, but let's not go down that rabbit hole for now.
To find the maximum number of students in the art class, we need to determine the possible dimensions of the rectangular solid that can be formed using the 48 identical 1 cm cubes.
Since each cube has a length of 1 cm, we can consider the factors of 48 to find the possible dimensions. The factors of 48 are: 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48.
Now, let's check which of these dimensions can actually be formed using the 48 cubes by using an equation.
Let's consider the length, width, and height of the rectangular solid as l, w, and h, respectively. According to the problem, volume of the solid equals 48:
l * w * h = 48
Now, let's try out the possible combinations of l, w, and h to determine if any of them satisfy the equation.
For each combination of values, if we can't make a rectangular solid with 48 cubes, we eliminate it as a possibility. Otherwise, we note down the combination.
After checking all the combinations, we find that the following combinations satisfy the equation:
1 * 2 * 24 = 48
1 * 3 * 16 = 48
1 * 4 * 12 = 48
1 * 6 * 8 = 48
2 * 3 * 8 = 48
2 * 4 * 6 = 48
3 * 4 * 4 = 48
So, there are 7 different combinations that satisfy the equation. Since each combination represents a unique rectangular solid, the maximum number of students in the art class is 7.