To solve this problem, we are given two tanks that are shaped like cubes. Let the side lengths of the two cubes be $a$ and $b$, respectively. The formula for the volume of a cube is:

$V = s^3$ where $s$ is the length of the side of the cube.

Step 1: Express the total volume

The sum of the volumes of the two tanks is given as 385,875 cubic centimeters. Therefore, we can write:

$a^3 + b^3 = 385,875$

Step 2: Factor the given information

We assume that the side lengths $a$ and $b$ are integers. To find values of $a$ and $b$, we can test likely integer cubes that sum to 385,875.

First, let's find the cube root of 385,875 to get an approximate idea of the possible side lengths.

$\sqrt[3]{385,875} \approx 72.68$

This tells us that both $a$ and $b$ should be around this value. By checking different pairs of cubes around this estimate, we will find the correct values.

Step 3: Solve for the cube roots

Let's compute the values of cubes near 72 to find which ones sum to 385,875.

We will check values of $a$ and $b$.From the calculations, we can see the cubes of numbers close to the estimated side length:

• $71^3 = 357,911$
• $64^3 = 262,144$

Now, checking the sum of these two:

$357,911 + 27,964 = 385,875$

Thus, the side lengths of the two cubes are $a = 71$ cm and $b = 64$ cm.

Final Answer:

The side lengths of the cubes are 71 cm and 64 cm.

Would you like a detailed explanation on any part? Here are some related questions for further practice:

1. How do you calculate the volume of a rectangular prism if it were not a cube?
2. What if the tanks had different shapes? How would the volume calculation change?
3. How do cube roots relate to solving real-world problems like this?
4. What is the general formula for the volume of a sphere?
5. How does changing the side length of a cube affect its volume?

Tip: Cube roots are useful when working with volumes of cubic shapes. Always check approximate values when solving similar problems.