you construct an open box with a square base (see figure) from 189 square inches of material. The height of the box is 3 inches.

A rectangular prism has a square base and the top face removed. The length is labeled x, the width is labeled x and the height is labeled 3 in.
What are the dimensions of the box in inches? (Hint: The surface area is S = x2 + 4xh.)

To solve this problem, we need to use the given surface area formula and the given dimensions of the box to find the value of \(x\), which is the length of the sides of the square base.

### Given Information:
- The surface area of the material used is 189 square inches.
- The height of the box is 3 inches.
- The formula for the surface area is:

\[
S = x^2 + 4xh
\]

where:
- \(S\) is the surface area,
- \(x\) is the length of one side of the square base,
- \(h\) is the height of the box.

Since the height of the box is 3 inches, we can substitute \(h = 3\) and \(S = 189\) into the formula.

\[
189 = x^2 + 4x(3)
\]

Simplifying the equation:

\[
189 = x^2 + 12x
\]

Now, we solve this quadratic equation for \(x\).

### Step 1: Rearrange the equation
\[
x^2 + 12x - 189 = 0
\]

### Step 2: Solve the quadratic equation

We can use the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For this equation, \(a = 1\), \(b = 12\), and \(c = -189\).

Now, let's calculate the value of \(x\).The two solutions for \(x\) are:

- \(x_1 = 9\)
- \(x_2 = -21\)

Since the length of the side of the base \(x\) must be positive, we discard the negative solution. Therefore, the value of \(x\) is 9 inches.

### Dimensions of the box:
- The side length of the square base is \(9\) inches.
- The height of the box is \(3\) inches.

Thus, the dimensions of the box are:
- \(9 \times 9 \times 3\) inches.

Would you like more details on the solution process, or do you have any questions?

### Related Questions:
1. How would the formula change if the height of the box were different?
2. How can you calculate the volume of this open box?
3. What would the surface area be if the height increased to 5 inches?
4. How can you solve the quadratic equation using factoring instead of the quadratic formula?
5. How would the dimensions change if the surface area increased to 250 square inches?

### Tip:
Always double-check units when solving geometry problems to ensure you're calculating areas, volumes, and dimensions in the correct units.

You construct an open box with a square base from 189 square inches of material. The height of the box is 3 inches. What are the dimensions of the box in inches? (Hint: The surface area is S = x^2 + 4xh.)

This guide explains how to find the dimensions of an open box with a square base using 189 square inches of material and a height of 3 inches. It involves solving a quadratic equation derived from the surface area formula. Suitable for students in Grades 8-10.

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