At a glassware factory, molten cobalt glass is poured into molds to make paperweights. Each mold is a rectanglar prism whose height is 3 inches greater than the length of each side of the square base. A machine pours 20 cubic inches of liquid glass into each mold. What are the dimensions of the mold?

Let's start by assigning variables to the dimensions of the mold.

Let's say the length of each side of the square base is "x" inches.

According to the question, the height is 3 inches greater than the length of each side of the square base. Therefore, the height is (x + 3) inches.

The volume of a rectangular prism can be calculated by multiplying the length, width, and height. In this case, the volume is given as 20 cubic inches.

So, we have the equation:

(x) * (x) * (x + 3) = 20

Now, let's solve this equation step-by-step to find the dimensions of the mold.

1. Expand the equation:
x^2 * (x + 3) = 20

2. Distribute:
x^3 + 3x^2 = 20

3. Subtract 20 from both sides:
x^3 + 3x^2 - 20 = 0

4. Factor the equation:
(x - 2)(x^2 + 5x + 10) = 0

The quadratic factor (x^2 + 5x + 10) cannot be factored further, so we'll focus on the linear factor (x - 2).

5. Set (x - 2) = 0 and solve for x:
x - 2 = 0
x = 2

So, the length of each side of the square base is 2 inches.

Now, let's find the height of the mold:

height = x + 3
height = 2 + 3
height = 5 inches

Therefore, the dimensions of the mold are a square base with sides measuring 2 inches and a height of 5 inches.

To find the dimensions of the mold, we need to set up an equation using the given information.

Let's assume the length of each side of the square base of the mold is "x".

According to the problem, the height of the mold is 3 inches greater than the length of each side of the square base. Therefore, the height of the mold is x + 3.

The volume of a rectangular prism is given by the formula: volume = length × width × height.

In this case, the length and width are both equal to x (since we have a square base). So, the volume of the mold is:

volume = x × x × (x + 3) = 20 cubic inches

To solve this equation, we can rewrite it as a quadratic equation:

x^2(x + 3) = 20

Expanding the equation:

x^3 + 3x^2 = 20

Rearranging the equation and bringing everything to one side:

x^3 + 3x^2 - 20 = 0

To solve this cubic equation, you can use numerical methods or a graphing calculator. One possible solution is x ≈ 2.658.

So, the length of each side of the square base is approximately 2.658 inches.

To find the height of the mold, we add 3 inches to the length of the base:

Height = x + 3 = 2.658 + 3 = 5.658 inches.

Therefore, the dimensions of the mold are approximately 2.658 inches for each side of the square base and 5.658 inches for the height.

volume=b^2 *( b+3)

20=b^3+3b^2
I think I would graph this:
b^3+3b^2-20=0

I don't see an easy solution to the cubic equation.