Elizabeth and Lillian have cube shaped bins to store their items on shelves. The volume of each storage cube is 415 cubic inches. The length of each side of the cube must be between 7 and 8 inches. Elizabeth thinks the length of the side of the cube is closer to 7 inches and Lillian thinks the length of the side of the cube is closer to 8 inches. Who is correct?
To determine which person is correct, we need to calculate the actual length of each side of the cube.
Let's first find the volume of the cube using the given volume of 415 cubic inches. The formula for the volume of a cube is V = s^3, where s is the length of each side.
415 = s^3
To solve for s, we take the cube root of both sides:
s = ∛(415)
Using a calculator, we find that s ≈ 7.91 inches.
Elizabeth estimated the length to be closer to 7 inches, while Lillian estimated it to be closer to 8 inches. Since 7.91 is closer to 8 than 7, Lillian is correct.
To determine who is correct, we need to find the length of the side of the cube that has a volume closest to 415 cubic inches.
First, let's calculate the volume of a cube with a side length of 7 inches:
Volume = Side length x Side length x Side length = 7 x 7 x 7 = 343 cubic inches.
Now, let's calculate the volume of a cube with a side length of 8 inches:
Volume = Side length x Side length x Side length = 8 x 8 x 8 = 512 cubic inches.
Out of these two volumes, neither is equal to 415 cubic inches. However, we can determine which one is closer by finding the difference between each volume and 415.
For a side length of 7 inches:
Difference = |Volume - 415| = |343 - 415| = 72 cubic inches.
For a side length of 8 inches:
Difference = |Volume - 415| = |512 - 415| = 97 cubic inches.
Comparing the differences, we can see that a cube with a side length of 7 inches is closer to 415 cubic inches than a cube with a side length of 8 inches. Therefore, Elizabeth is correct in thinking that the length of the side of the cube is closer to 7 inches.
To determine who is correct, we need to calculate the volume of the cube using both the lower and upper bounds for the length of each side.
First, let's calculate the volume of the cube assuming the length of each side is 7 inches:
Volume = Length x Width x Height
Volume = 7 inches x 7 inches x 7 inches
Volume = 343 cubic inches
Next, let's calculate the volume of the cube assuming the length of each side is 8 inches:
Volume = Length x Width x Height
Volume = 8 inches x 8 inches x 8 inches
Volume = 512 cubic inches
Now, we compare the calculated volumes with the given volume of 415 cubic inches:
- 343 cubic inches is less than 415 cubic inches.
- 512 cubic inches is greater than 415 cubic inches.
Based on these calculations, we can conclude that Elizabeth is correct. The length of the side of the cube is closer to 7 inches.