what are the dimentions of three different rectangular prisms that each have a surface area of 72?
The small factors of 72 are 1,2,3,4,6,8
pick any of those and calculate the other by dividing it into 72.
Ah, three rectangular prisms with a surface area of 72? That's quite intriguing! Let me show you three boxes that are determined to be the life of the party:
1) The "Smarty Pants" Prism:
Height = 2 units, Width = 6 units, Length = 6 units.
2) The "Squats-A-Lot" Prism:
Height = 3 units, Width = 4 units, Length = 6 units.
3) The "Stretch Armstrong" Prism:
Height = 1 unit, Width = 8 units, Length = 9 units.
Remember, these dimensions are just for fun, but they will surely bring out the laughter at any geometry party!
To find the dimensions of three different rectangular prisms with a surface area of 72, you need to consider the factors of 72. The factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72.
Let's start by finding one possible set of dimensions:
1. Length: 4 units, Width: 6 units, Height: 3 units
To calculate the surface area, use the formula: Surface Area = 2lw + 2lh + 2wh
Surface Area = 2 * 4 * 6 + 2 * 4 * 3 + 2 * 6 * 3
Surface Area = 48 + 24 + 36
Surface Area = 108
Since the surface area of the prism with dimensions 4 x 6 x 3 is greater than 72, this set of dimensions does not satisfy the requirement.
Now, let's find another set of dimensions:
2. Length: 2 units, Width: 12 units, Height: 3 units
Using the same formula: Surface Area = 2lw + 2lh + 2wh
Surface Area = 2 * 2 * 12 + 2 * 2 * 3 + 2 * 12 * 3
Surface Area = 48 + 12 + 72
Surface Area = 132
Since this surface area is greater than 72, this set of dimensions also doesn't satisfy the requirement.
Finally, let's find the last set of dimensions:
3. Length: 3 units, Width: 4 units, Height: 6 units
Surface Area = 2 * 3 * 4 + 2 * 3 * 6 + 2 * 4 * 6
Surface Area = 24 + 36 + 48
Surface Area = 108
This set of dimensions satisfies the requirement, as the surface area is equal to 72.
Therefore, one possible set of dimensions for a rectangular prism with a surface area of 72 is 3 x 4 x 6 units.
To find the dimensions of three different rectangular prisms with the same surface area of 72, we can use a formula that relates the surface area of a rectangular prism to its dimensions. The formula for the surface area of a rectangular prism is:
Surface Area = 2*(Length * Width + Length * Height + Width * Height)
Given that the surface area is 72, we can set up an equation:
72 = 2*(Length * Width + Length * Height + Width * Height)
Now, we need to find three different sets of dimensions that satisfy this equation.
One approach is to start by assuming different values for one of the dimensions, and then solving for the remaining dimensions. Let's assume the length of the rectangular prism is 6:
72 = 2*(6 * Width + 6 * Height + Width * Height)
Simplifying the equation, we have:
36 = Width * Height + 6 * (Width + Height)
Now, we can try different values for Width and Height that satisfy this equation. For example, if Width = 2 and Height = 4, the equation becomes:
36 = 2 * 4 + 6 * (2 + 4)
36 = 8 + 6 * 6
36 = 8 + 36
This set of dimensions (Length = 6, Width = 2, Height = 4) satisfies the equation and has a surface area of 72.
To find two more sets of dimensions, you can repeat this process by assuming different values for the length and solving for the remaining dimensions. For example, if Length = 4:
72 = 2*(4 * Width + 4 * Height + Width * Height)
36 = 2 * Width + 2 * Height + Width * Height
You can then explore different combinations of Width and Height that satisfy this equation. Repeat this process one more time with a different assumed value for the length to find the third set of dimensions.
In summary, to find three different rectangular prisms with a surface area of 72, you can use the formula for surface area and solve the resulting equation for different sets of dimensions by assuming different values for one of the dimensions and then finding the remaining dimensions.