Let's solve each of these problems one by one:

Problem 7: Benjamin rents a storage unit shaped like a cube. There are 12 identical storage units in each row. Each unit has a volume of 125 cubic feet. What is the length of each row?

• Step 1: Find the side length of one cube. The volume of a cube $V$ is given by $V = s^3$, where $s$ is the side length.

  $125 = s^3$

  Take the cube root of both sides:

  $s = \sqrt[3]{125} = 5 \, \text{feet}$

• Step 2: Multiply the side length by 12 to find the length of each row (since there are 12 storage units in a row):

  $\text{Length of row} = 12 \times 5 = 60 \, \text{feet}$

Problem 8: Would you classify the number 55 as a perfect square, a perfect cube, both, or neither?

• A perfect square is a number that can be written as $n^2$ for some integer $n$.

  • The square root of 55 is about 7.42, which is not an integer.

• A perfect cube is a number that can be written as $n^3$ for some integer $n$.

  • The cube root of 55 is about 3.83, which is also not an integer.

Thus, 55 is neither a perfect square nor a perfect cube.

Problem 9: Critique Reasoning: Clara says that if you square the number 4 and then divide the result by 2, you end up with 4. Is Clara correct?

• Step 1: Square the number 4.

  $4^2 = 16$

• Step 2: Divide by 2.

  $\frac{16}{2} = 8$

Clara is incorrect. The result is 8, not 4.

Problem 10: A fish tank has a volume of 1,568 cubic feet and a depth of 8 feet. If the base of the tank is square, what is the length of each side of the tank?

• Step 1: The volume of the tank is given by:

  $\text{Volume} = \text{Base Area} \times \text{Height}$

  The base area is a square, so let the side of the square be $s$. Then:

  $1568 = s^2 \times 8$

• Step 2: Solve for $s^2$:

  $s^2 = \frac{1568}{8} = 196$

• Step 3: Find $s$ by taking the square root:

  $s = \sqrt{196} = 14 \, \text{feet}$

Would you like further details on any of these problems? Here are five related questions to consider:

1. How would you solve problem 7 if the storage units were not cubes?
2. Can you find another number that is neither a perfect square nor a perfect cube but close to 55?
3. What happens if Clara divides by a different number, say 4, instead of 2? Does her statement hold then?
4. In problem 10, what would happen if the depth was halved while keeping the same volume?
5. Can you think of any practical examples where volume calculations, like in problem 10, are useful?

Tip: Always check whether a number is a perfect square by finding the nearest integers before and after its square root.