To solve this problem, we will use the surface area formula for an open box with a square base. The surface area of the box consists of the base and the four sides (since the top is removed). Let's break it down step by step.

Variables:

• The length of the base = $x$ inches
• The width of the base = $x$ inches (because the base is square)
• The height of the box = 3 inches

Surface area formula:

The surface area $S$ of the box is the area of the base plus the area of the four sides: $S = x^2 + 4(x \cdot h)$ Where:

• $x^2$ is the area of the square base
• $4(x \cdot h)$ is the total area of the four vertical sides (height = $h = 3$ inches)

Given that the surface area is 189 square inches, we can set up the equation: $189 = x^2 + 4(x \cdot 3)$ Simplify: $189 = x^2 + 12x$

Solve for $x$:

This is a quadratic equation: $x^2 + 12x - 189 = 0$ We will solve this using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Where $a = 1$, $b = 12$, and $c = -189$.

Let's calculate it step by step.The solutions to the quadratic equation are $x = 9$ and $x = -21$. Since the dimension of the base length $x$ cannot be negative, we select $x = 9$ inches.

Final dimensions of the box:

• Length = 9 inches
• Width = 9 inches (since the base is square)
• Height = 3 inches

Would you like more details on any part of the solution, or do you have any other questions?

Here are 5 related questions for practice:

1. How would the dimensions change if the surface area were 300 square inches?
2. What would the height of the box be if the base side length remained 9 inches but the surface area increased to 250 square inches?
3. What is the volume of the open box?
4. If the top of the box were added, how would the surface area change?
5. How does changing the height affect the total surface area for a fixed base size?

Tip: When solving quadratic equations, always check for physical constraints like positive lengths when selecting a solution.