https://file.aiquickdraw.com/mathbot/file/1730256565395b4j3g419.jpg

To tackle this problem step-by-step, let's go through each part of the exercise based on the information provided:

---

### Problem Analysis
1. We start with a square piece of cardboard with a width of 3 feet.
2. A square with side length \( x \) is cut from each corner, and the sides are bent up to form an open-top box.

Given this setup, let:
- \( x \): Side length of the squares being cut out.
- \( y \): Length of the base of the box after folding.

Now, let's solve each part in turn.

---

### Solution Steps

#### Part (a): Diagram and Volume Estimation
You would draw various configurations of the box by changing \( x \), the side of the square being cut out. For each \( x \), calculate the volume \( V = y \times y \times x \), where \( y \) is the length of the base after cutting out the corners.

#### Part (b): General Diagram
Draw a square with a smaller square (of side \( x \)) cut out at each corner. After folding, the height of the box becomes \( x \), and the base length \( y \) of the box becomes \( 3 - 2x \), as cutting \( x \) from both sides reduces the width by \( 2x \).

#### Part (c): Expression for Volume in Terms of \( x \) and \( y \)
Since the volume \( V \) of a box is given by:
\[
V = \text{base area} \times \text{height}
\]
we have:
\[
V = y \times y \times x = y^2 \cdot x
\]
Thus, the volume \( V \) can be expressed as:
\[
V = x \cdot y^2
\]

#### Part (d): Relationship Between \( x \) and \( y \)
Since \( y \) represents the length of the base after cutting, and we start with a 3-foot wide piece of cardboard, we have:
\[
y = 3 - 2x
\]
This equation expresses \( y \) in terms of \( x \).

#### Part (e): Volume as a Function of \( x \) Only
Substitute \( y = 3 - 2x \) into the volume equation:
\[
V(x) = x \cdot (3 - 2x)^2
\]
Expanding \( (3 - 2x)^2 \) gives:
\[
V(x) = x \cdot (9 - 12x + 4x^2)
\]
\[
V(x) = 4x^3 - 12x^2 + 9x
\]
So, the volume as a function of \( x \) is:
\[
V(x) = 4x^3 - 12x^2 + 9x
\]

#### Part (f): Maximizing the Volume
To find the maximum volume, take the derivative \( V'(x) \) and set it to zero:
1. Differentiate \( V(x) = 4x^3 - 12x^2 + 9x \).
2. Solve \( V'(x) = 0 \) for \( x \) to find critical points.
3. Substitute back to find the maximum volume.

---

Would you like a detailed breakdown of finding the maximum volume in Part (f) using calculus? 

---

### Follow-Up Questions

1. How does changing the value of \( x \) affect the height and base area of the box?
2. What are the physical limitations on the value of \( x \) for this box construction?
3. How do we verify that a critical point provides a maximum volume and not a minimum?
4. What would happen to the maximum volume if the cardboard’s width changed from 3 feet to a different size?
5. Can you think of a real-world application for maximizing the volume of an open-top box in design or packaging?

---

### Tip
When solving optimization problems, always check the domain of possible values for variables to avoid unrealistic results.

Consider the following problem: a box with an open top is to be constructed from a square piece of cardboard, 3 feet wide, by cutting out a square from each of the four corners and bending up the sides. Find the largest volume that such a box can have.

Learn how to maximize the volume of an open-top box created by cutting corners from a 3-foot square cardboard. This optimization problem involves volume calculation, derivatives, and algebra, making it ideal for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation