To solve this problem, let's break it down step by step.

Problem Restatement

Raphael is building a stone landscape around a sculpture, which measures 6 meters by 10 meters. The stone landscape should have uniform width $x$ and wrap around the two short sides and two of the long sides of the sculpture. We need to write an equation for the area of the stone landscape.

Step 1: Define the dimensions

• Sculpture dimensions: 6 meters by 10 meters.
• Width of the landscape: $x$ meters.

Step 2: Total dimensions with the landscape

The landscape wraps around the sculpture, so the total dimensions including the landscape will be:

• Length: $10 + 2x$ (adding $x$ on both sides of the length).
• Width: $6 + 2x$ (adding $x$ on both sides of the width).

Step 3: Calculate the total area

The area of the entire rectangle (sculpture + landscape) is given by: $\text{Total Area} = (\text{Length}) \times (\text{Width}) = (10 + 2x) \times (6 + 2x)$

Step 4: Calculate the area of the sculpture

The area of the sculpture itself is: $\text{Sculpture Area} = 10 \times 6 = 60 \text{ square meters}$

Step 5: Find the area of the landscape

The area of the landscape alone will be the total area minus the area of the sculpture: $\text{Landscape Area} = \text{Total Area} - \text{Sculpture Area} = (10 + 2x)(6 + 2x) - 60$

Final Equation

$\text{Landscape Area} = (10 + 2x)(6 + 2x) - 60$

Would you like to see the expansion of this equation or further steps?

Related Questions

1. What happens to the landscape area if the width $x$ is doubled?
2. How does the landscape area change if the sculpture was a square instead?
3. Can you calculate the exact landscape area if $x = 2$ meters?
4. What would the equation look like if the landscape wrapped around all four sides?
5. How does changing the dimensions of the sculpture affect the landscape area equation?

Tip

When solving word problems involving geometric shapes, it’s often helpful to draw a diagram to visualize the problem better and ensure that all components are accounted for.