https://file.aiquickdraw.com/mathbot/file/1725421723846a1grp21s.jpg

The problem in the image involves determining the polynomial that represents the volume of a photo frame. Here's how to approach the problem:

### Problem Description
- The frame is rectangular, with dimensions provided in the diagram: the width is \( 25 \, \text{cm} \) and the height is \( 30 \, \text{cm} \).
- A square section with a side length of \( x \, \text{cm} \) is cut from each corner of the frame, and the edges are then folded up to form a box.

### Goal
We need to determine the polynomial that correctly represents the volume \( V(x) \) of this box.

### Steps to Solve:

1. **Determine the dimensions of the box after folding**:
   - The original width of the frame is \( 25 \, \text{cm} \), but after removing \( x \) cm from both sides, the width becomes \( (25 - 2x) \, \text{cm} \).
   - Similarly, the original height of the frame is \( 30 \, \text{cm} \), so after removing \( x \) cm from both sides, the height becomes \( (30 - 2x) \, \text{cm} \).
   - The height of the box, after folding the edges up, is \( x \, \text{cm} \).

2. **Volume Formula**:
   The volume \( V(x) \) of the box is given by the product of its length, width, and height:
   \[
   V(x) = \text{length} \times \text{width} \times \text{height}
   \]
   Plugging in the values:
   \[
   V(x) = x \times (25 - 2x) \times (30 - 2x)
   \]

3. **Expand and Simplify**:
   To find the polynomial, expand the expression:
   \[
   V(x) = x \times \left[(25 - 2x)(30 - 2x)\right]
   \]
   First, expand the product of the binomials:
   \[
   (25 - 2x)(30 - 2x) = 750 - 50x - 60x + 4x^2 = 750 - 110x + 4x^2
   \]
   Now multiply by \( x \):
   \[
   V(x) = x \times (750 - 110x + 4x^2) = 4x^3 - 110x^2 + 750x
   \]

4. **Identify the Correct Option**:
   Comparing this with the given options, the correct polynomial is:

   \[
   \boxed{\text{Option a) } V(x) = 4x^3 - 110x^2 + 750x}
   \]

Would you like further details or have any other questions?

### Related Questions:
1. What happens to the volume if \( x \) is increased slightly?
2. How would you find the maximum possible volume of the box?
3. What are the dimensions of the box if \( x = 5 \, \text{cm} \)?
4. How does the volume change if you only change the width or height of the original frame?
5. How would you verify that \( V(x) \) is a cubic polynomial?

### Tip:
When dealing with problems that involve cutting and folding, carefully track how dimensions change at each step. This is key to setting up the correct polynomial for volume or area.

Sebuah perusahaan akan membuat produk figura foto eksklusif. Figura foto tersebut berbentuk dari lembaran besi stainless dengan ketebalan 2 mm yang dipotong bagian tengahnya. Bagian pinggir yang tersisa akan dilipat untuk membentuk kotak. Tentukan fungsi polinomial yang menyatakan volume figura foto tersebut.

This problem involves calculating the volume of a rectangular box by using polynomial expansion. The problem requires understanding of algebraic principles, including the expansion and simplification of polynomials. Suitable for 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