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.)

Incorrect: Your answer is incorrect.
in. (length) ✕
Correct: Your answer is correct.
in. (width) ✕
Correct: Your answer is correct.
in. (height)

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.

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. What are the dimensions of the box in inches?

This guide walks you through solving a geometry problem involving an open box with a square base. Using 189 square inches of material and a height of 3 inches, learn how to apply quadratic equations to find the dimensions of the box. Suitable for students in Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation