Math Problem Statement

Suppose that the volume of the box is 182 cubic centimeters. Use the given information to write a quadratic equation that can be used to find x.

Solution

In this problem, we are given the dimensions of an open box, and the task is to write a quadratic equation to find the unknown dimension xx, given that the volume of the box is 182 cubic centimeters.

Step 1: Understand the box dimensions

  • The length of the base is x+6x + 6.
  • The width of the base is xx.
  • The height of the box is 2 cm (as shown by the folded sides).

Step 2: Write the formula for the volume of a rectangular box

The formula for the volume VV of a rectangular box is:

V=length×width×heightV = \text{length} \times \text{width} \times \text{height}

Using the dimensions from the diagram:

  • Length = x+6x + 6
  • Width = xx
  • Height = 22

The volume equation becomes:

V=(x+6)x2V = (x + 6) \cdot x \cdot 2

Step 3: Substitute the given volume

We are told that the volume is 182 cubic centimeters, so:

(x+6)x2=182(x + 6) \cdot x \cdot 2 = 182

Step 4: Simplify the equation

First, divide both sides by 2 to simplify:

(x+6)x=91(x + 6) \cdot x = 91

Now, expand the left-hand side:

x(x+6)=91x(x + 6) = 91 x2+6x=91x^2 + 6x = 91

Step 5: Form the quadratic equation

Rearrange the equation to standard quadratic form:

x2+6x91=0x^2 + 6x - 91 = 0

This is the quadratic equation that can be used to solve for xx.

Would you like details on solving this quadratic equation, or any other questions on this topic?

5 Related Questions:

  1. How can you solve the quadratic equation x2+6x91=0x^2 + 6x - 91 = 0?
  2. What are the possible values of xx once the quadratic equation is solved?
  3. How does the box's volume change if the height is doubled?
  4. What is the relationship between the volume and the surface area of the box?
  5. Can you use the quadratic formula to solve this type of box dimension problem?

Tip:

When expanding or simplifying expressions, always check for common factors to reduce the equation before solving. This helps avoid mistakes and simplifies the process!

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Equations
Volume of Rectangular Prisms

Formulas

Volume of a rectangular box: V = length × width × height
Standard form of a quadratic equation: ax^2 + bx + c = 0

Theorems

Expansion of binomials
Rearranging equations to standard form

Suitable Grade Level

Grades 8-10