the edge of one cube is 2 cm shorter than the edge of a second cube. The volumes of the two cubes differ by 386cm^3. Find the edge of the smaller cube.

(x+2)^3 - x^3 = 386

6x^2 + 12x + 8 = 386
x^2 + 2x - 63
(x+9)(x-7)

the shorter side is 7

check: 9^3-7^3 = 729-343 = 386

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

The edge length of the smaller cube is then (x - 2) cm.

The volume of a cube is given by the formula V = s^3, where V is the volume and s is the length of the side.

The volume of the larger cube is x^3 cm^3.
The volume of the smaller cube is (x - 2)^3 cm^3.

According to the problem, the difference in volume is 386 cm^3.

So, we can write the equation as:

x^3 - (x - 2)^3 = 386

Expanding (x - 2)^3, we get:

x^3 - (x^3 - 6x^2 + 12x - 8) = 386

By simplifying, we have:

x^3 - x^3 + 6x^2 - 12x + 8 = 386

Rearranging the equation, we get:

6x^2 - 12x + 8 - 386 = 0

Combining like terms, we have:

6x^2 - 12x - 378 = 0

Dividing the equation by 6 to simplify, we get:

x^2 - 2x - 63 = 0

To solve this quadratic equation, we can factorize or use the quadratic formula. Factoring is simpler in this case. We look for two numbers whose product is -63 and whose sum is -2.

After some calculation, we find the factors: (x - 9)(x + 7) = 0.

Setting each factor equal to zero, we get x - 9 = 0 or x + 7 = 0.

Solving these equations, we find x = 9 or x = -7.

Since the length of a side cannot be negative, we discard x = -7.

Therefore, the edge of the larger cube is 9 cm.

The edge of the smaller cube is (9 - 2) = 7 cm.

So, the edge of the smaller cube is 7 cm.

To solve this problem, we can set up equations using the information given.

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

According to the problem, the edge length of the larger cube is 2 cm longer than the edge length of the smaller cube. This means the edge length of the larger cube is "x + 2" cm.

The volume of a cube is given by the formula V = (side length)^3.

So, the volume of the smaller cube is V1 = x^3 cm^3, and the volume of the larger cube is V2 = (x + 2)^3 cm^3.

We are also given that the volumes of the two cubes differ by 386 cm^3. So, we can set up the following equation:

V2 - V1 = 386 cm^3

Substituting the expressions for V1 and V2, we have:

(x + 2)^3 - x^3 = 386

Expanding the left side of the equation:

x^3 + 6x^2 + 12x + 8 - x^3 = 386

Simplifying the equation:

6x^2 + 12x + 8 = 386

Rearranging the equation:

6x^2 + 12x + 8 - 386 = 0

Combining like terms:

6x^2 + 12x - 378 = 0

Now, we can solve this quadratic equation for x by factoring, completing the square, or using the quadratic formula.

Let's use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = 6, b = 12, and c = -378. Substituting these values in the quadratic formula, we have:

x = (-12 ± √((12)^2 - 4 * 6 * -378)) / (2 * 6)

Simplifying further:

x = (-12 ± √(144 + 912)) / 12

x = (-12 ± √1056) / 12

Now we can simplify the square root:

x = (-12 ± 32.5) / 12

x = (-12 + 32.5) / 12 or x = (-12 - 32.5) / 12

x = 20.5 / 12 or x = -44.5 / 12

x = 1.708 or x = -3.708

Since the edge length of a cube cannot be negative, we disregard x = -3.708.

Therefore, the edge length of the smaller cube is approximately 1.708 cm.