The edge of one cube is 5cm shorter than the edge of a second cube. The volumes of the two cubes differ by 875cm^3. Find the edge of the smaller cube.

Let's assume that the edge of the smaller cube is 'x' cm.

According to the given information, the edge of the larger cube is 5 cm longer than the edge of the smaller cube. Thus, the edge of the larger cube would be (x + 5) cm.

The volume of a cube is calculated by raising the length of one of its sides to the power of 3. Therefore, the volume of the smaller cube would be x^3 cm^3, and the volume of the larger cube would be (x + 5)^3 cm^3.

Given that the volumes of the two cubes differ by 875 cm^3, we can set up the following equation:

(x + 5)^3 - x^3 = 875

Expanding the equation, we get:

x^3 + 3x^2 * 5 + 3x * (5^2) + 5^3 - x^3 = 875

Further simplifying the equation, we can remove the x^3 terms:

3x^2 * 5 + 3x * (5^2) + 5^3 = 875

Next, we can solve the equation for x.

Multiply out the terms:

15x^2 + 75x + 125 = 875

Subtract 875 from both sides to isolate the x terms:

15x^2 + 75x + 125 - 875 = 0

15x^2 + 75x - 750 = 0

Finally, we can divide the equation by 15 to simplify it further:

x^2 + 5x - 50 = 0

Now, we have a quadratic equation that we can solve using factoring, completing the square, or using the quadratic formula. In this case, let's use factoring:

(x + 10)(x - 5) = 0

This means that x + 10 = 0 or x - 5 = 0.

If x + 10 = 0, then x = -10. However, since we're dealing with lengths, a negative value is not possible, so we can ignore this solution.

If x - 5 = 0, then x = 5.

Therefore, the edge of the smaller cube is 5 cm.

To find the edge of the smaller cube, we can use the following steps:

Let's assume that the edge length of the larger cube is x cm.

According to the problem, the edge length of the smaller cube is 5 cm shorter than the edge length of the larger cube. So the edge length of the smaller cube can be expressed as x - 5 cm.

The volume of a cube can be calculated using the formula V = s^3, where V represents volume and s represents the length of the edge.

So, the volume of the larger cube is V1 = x^3.

The volume of the smaller cube is V2 = (x - 5)^3.

Given that the difference between the volumes of the two cubes is 875 cm^3, we can set up the equation:

V1 - V2 = 875

(x^3) - ((x - 5)^3) = 875

Expanding the binomial (x - 5)^3, we get:

x^3 - 3x^2 * 5 + 3x * 5^2 - 5^3 = 875

x^3 - 3x^2 * 5 + 3x * 25 - 125 = 875

x^3 - 15x^2 + 75x - 125 = 875

Rearranging the equation, we have:

x^3 - 15x^2 + 75x - 1000 = 0

Now, we can solve this equation to find the value of x (the edge length of the larger cube) using numerical methods or a calculator.

Using numerical methods, we find that the value of x is approximately 11.83 cm.

Finally, to find the edge length of the smaller cube, we substitute this value back into the expression x - 5:

11.83 cm - 5 cm = 6.83 cm.

Therefore, the edge length of the smaller cube is approximately 6.83 cm.