The edge of one cube is 2 cm shorter than the edge of a second cube. The volumes of the two cubes differ by 386 cm^3. Find the edge of the smaller cube using factoring.
To find the edge of the smaller cube, we can use factoring. Let's assume that the edge of the second cube is x cm.
According to the information given, the edge of the first cube is 2 cm shorter than the edge of the second cube. This means that the edge of the first cube is (x - 2) cm.
The volume of a cube is calculated by cubing its edge. Therefore, the volume of the first cube is (x - 2)^3 cm^3, and the volume of the second cube is x^3 cm^3.
We are told that the volumes of the two cubes differ by 386 cm^3. This can be expressed as an equation:
x^3 - (x - 2)^3 = 386
To simplify this equation, we need to expand the cube on the right side:
x^3 - (x - 2)(x - 2)(x - 2) = 386
Expanding the cube on the right side, we get:
x^3 - (x^2 - 4x + 4x - 4)(x - 2) = 386
Simplifying further:
x^3 - (x^3 - 4x^2 + 8x - 4x - 8) = 386
Combining like terms:
x^3 - x^3 + 4x^2 - 8x + 4x + 8 = 386
Simplifying:
4x^2 - 4x + 8 = 386
Now we have a quadratic equation. Rearranging it to make it easier to solve:
4x^2 - 4x - 378 = 0
To solve this quadratic equation, we can factor it by finding two numbers that multiply to -378 and sum to -4. The numbers are -18 and 21.
So, factoring the quadratic equation:
4x^2 - 4x - 378 = (2x - 21)(2x + 18) = 0
Setting each factor equal to zero:
2x - 21 = 0 or 2x + 18 = 0
Solving for x:
2x = 21 or 2x = -18
x = 21/2 or x = -18/2
We discard the negative value since the edge of a cube cannot be negative. Therefore, x = 21/2.
Now, we can find the edge of the smaller cube by subtracting 2 cm from x:
Edge of the smaller cube = x - 2 = 21/2 - 2 = 21/2 - 4/2 = 17/2 cm
So, the edge of the smaller cube is 17/2 cm or 8.5 cm.