Question

crate a has a volume of 80 cubic feet. crate b has a volume of 64 cubic feet. youre trying to figure out which one is closer to having a side length of 4 feet. questions: 1. without using a calculator, how could you figure out the side lengths of these two cubes? 2. what math facts or patterns might help you estimate cube - shaped side lengths more easily?

Explanation:

Step1: Recall volume - side - length relationship

For a cube, the volume formula is $V = s^{3}$, where $V$ is the volume and $s$ is the side - length. To find the side - length, we need to find the cube - root of the volume, i.e., $s=\sqrt[3]{V}$.

Step2: Identify relevant math facts

Memorizing perfect - cube values helps. For example, $1^{3}=1$, $2^{3}=8$, $3^{3}=27$, $4^{3}=64$, $5^{3}=125$, etc. This helps in quickly estimating cube - shaped side - lengths.

Answer:

1. Since the volume formula for a cube is $V = s^{3}$, to find the side - length of a cube without a calculator, we need to find the number which, when multiplied by itself three times, gives the volume. In other words, we find the cube - root of the volume. For crate A with volume $V_A = 30$ cubic feet and crate B with volume $V_B=64$ cubic feet, we find the number $s$ such that $s^{3}=V$.
2. Memorizing perfect - cube values like $1^{3}=1$, $2^{3}=8$, $3^{3}=27$, $4^{3}=64$, $5^{3}=125$, etc. helps in estimating cube - shaped side - lengths. If the volume is close to a perfect - cube value, we can quickly estimate the side - length. For example, since crate B has a volume of 64 cubic feet and $4^{3}=64$, the side - length of crate B is 4 feet. For crate A with a volume of 30 cubic feet, since $3^{3}=27$ and $4^{3}=64$, the side - length of crate A is between 3 and 4 feet and closer to 3 feet.