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.