2. Squares of side a are cut from each corner of a 8 in x 6 in rectangle, so that its sides can be folded to make a box with no top. Represent a function in terms of a that can define the volume of the box. Answer in a simple context. 

To find a simple expression for the volume of a box made by cutting squares from the corners of an 8-inch by 6-inch rectangle and folding up the sides, we can follow these steps:

1. **Initial Rectangle Dimensions:**
   - Length: 8 inches
   - Width: 6 inches

2. **Squares Cut from Each Corner:**
   - Side of each square: \( a \)

3. **New Dimensions of the Box's Base:**
   - New length: \( 8 - 2a \)
   - New width: \( 6 - 2a \)

4. **Height of the Box:**
   - Height: \( a \) (the same as the side of the square cut out)

5. **Volume of the Box:**
   - Volume \( V \) = (length) \(\times\) (width) \(\times\) (height)
   - So, \( V(a) = (8 - 2a)(6 - 2a)a \)

This gives us the volume of the box as a function of \( a \):

\[
V(a) = (8 - 2a)(6 - 2a)a
\]

To summarize:

\[
V(a) = (8 - 2a)(6 - 2a)a
\]

This is the function that defines the volume of the box in terms of \( a \).

---

Would you like details or have any questions about this solution? Here are 8 related questions to further explore the topic:

1. What is the maximum possible value of \( a \) for which the box can be formed?
2. How does the volume change as \( a \) increases from 0 to its maximum value?
3. What is the volume of the box when \( a = 1 \) inch?
4. How can you find the value of \( a \) that gives the maximum volume of the box?
5. What are the dimensions of the box when \( a \) is at its maximum value for volume?
6. How do you determine if the volume function \( V(a) \) is correct?
7. Can you graph the volume function to visualize its behavior?
8. What practical applications can this volume function have in real-world scenarios?

**Tip:** To solve for the maximum volume, you can use calculus techniques, such as finding the derivative of the volume function \( V(a) \) and setting it to zero to find the critical points.

Learn how to calculate the volume of a box formed by cutting squares from the corners of an 8-inch by 6-inch rectangle and folding up the sides. This problem involves geometry and algebra, focusing on the volume formula of a box. Suitable for students in Grades 7-9.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation