https://r2.erweima.ai/mathbot/image/1726629552398k0hsvhio.jpg

The problem states that there is a poster with a base \( x \) and a height such that the total area is 540 cm². The poster includes margins of 2 cm on both sides and 1.5 cm on the top and bottom, and we are tasked with expressing the printed area in terms of the base of the poster.

Let’s break down the information:

- The total area of the poster is 540 cm².
- The margins reduce the printable area. Specifically, 2 cm is taken on each side, so the printable width becomes \( x - 4 \) cm (since \( 2 \) cm is taken on both the left and right sides).
- Similarly, the margins on the top and bottom are 1.5 cm each, so the printable height becomes \( h - 3 \) cm (since 1.5 cm is taken from both the top and bottom).

Now, we know the total area of the poster is:

\[
x \cdot h = 540 \, \text{cm}^2
\]

We want to express the printed area (without margins) as a function of \( x \).

### Step 1: Find the height in terms of \( x \)

From the total area equation, we can solve for the height:

\[
h = \frac{540}{x}
\]

### Step 2: Express the printed area

The printed area \( A_{\text{printed}} \) is given by the product of the printable width \( (x - 4) \) and the printable height \( \left( \frac{540}{x} - 3 \right) \):

\[
A_{\text{printed}} = (x - 4) \left( \frac{540}{x} - 3 \right)
\]

### Final Expression:

\[
A_{\text{printed}} = (x - 4) \left( \frac{540}{x} - 3 \right)
\]

This is the expression for the printed area in terms of the base \( x \).

---

Would you like to simplify this further or explore any other aspect of the problem?

Here are 5 questions that could extend this discussion:

1. How does the printed area change as the base \( x \) increases?
2. Can you find the value of \( x \) that maximizes the printed area?
3. What happens to the printed area if the margins are changed?
4. How would the formula for the printed area change if the total area of the poster were different?
5. Can you graph the function \( A_{\text{printed}}(x) \) to visualize its behavior?

**Tip:** When dealing with optimization problems in geometry, expressing variables in terms of one another helps simplify complex scenarios.

Un cartel de base x y altura y tiene un área de 540 cm² con márgenes de 2 cm a los lados y 1.5 cm en las partes superior e inferior. Expresa el área impresa en función de la base del cartel.

Learn how to express the printable area of a poster with a total area of 540 cm² and margins of 2 cm on the sides and 1.5 cm on the top and bottom, in terms of its base. The solution involves key concepts from algebra and geometry.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation