Framed Portrait Area: Expressing It In Terms Of Width
Let's dive into the world of portraits and frames, and figure out how to calculate the area of a framed portrait based on its width. This is a classic problem that combines geometry and algebra, making it a fun and practical exercise. We'll break it down step by step, ensuring you understand every part of the process. So, grab your thinking cap, and let's get started!
Understanding the Portrait Dimensions
First, let's clarify the dimensions of the portrait without the frame. The problem states that the height of the portrait is 1.5 times its width, which we'll denote as w. Therefore, the height h can be expressed as:
h = 1.5w
This relationship is crucial because it links the height and width, allowing us to express the area in terms of a single variable, w. The area of the portrait itself, without the frame, is simply the product of its width and height:
Area (portrait) = w * h* = w * (1.5w) = 1.5w^2
However, we're interested in the area of the framed portrait, so we need to consider the additional width provided by the frame.
Accounting for the Frame
The frame adds 3 inches to all sides of the portrait. This means that the width of the framed portrait is the original width plus twice the frame width (once on each side). Similarly, the height of the framed portrait is the original height plus twice the frame width. Let's calculate these new dimensions:
Framed width = w + 2 * 3 = w + 6
Framed height = 1.5w + 2 * 3 = 1.5w + 6
Now we have the dimensions of the entire framed portrait. The next step is to calculate the area of this framed portrait.
Calculating the Area of the Framed Portrait
To find the area of the framed portrait, we multiply the framed width by the framed height:
Area (framed) = (Framed width) * (Framed height) = (w + 6) * (1.5w + 6)
Now, let's expand this expression:
Area (framed) = w * (1.5w) + w * 6 + 6 * (1.5w) + 6 * 6
Area (framed) = 1.5w^2 + 6w + 9w + 36
Combine like terms:
Area (framed) = 1.5w^2 + 15w + 36
This expression represents the area of the framed portrait in terms of w. This is a quadratic expression, indicating that the area increases non-linearly as the width w increases. The w^2 term dominates when w is large, but the linear and constant terms also contribute significantly, especially for smaller values of w.
Putting it All Together
So, to recap, we started with the dimensions of the portrait, accounted for the frame, and then calculated the area of the framed portrait. The final expression for the area of the framed portrait in terms of w is:
Area (framed) = 1.5w^2 + 15w + 36
This comprehensive approach ensures that you understand not only the final answer but also the reasoning and calculations behind it. Understanding each step allows you to apply these techniques to similar problems in the future.
Common Mistakes to Avoid
When solving problems like this, there are a few common mistakes to watch out for:
- Forgetting to account for the frame on both sides: Remember that the frame increases the width and height by 3 inches on each side, so you need to add 6 inches to both dimensions.
- Incorrectly expanding the expression: Be careful when multiplying out the terms in the expression (w + 6) * (1.5w + 6). Ensure you distribute each term correctly.
- Not combining like terms: After expanding the expression, make sure to combine the w terms to simplify the final expression.
Avoiding these common mistakes will help you arrive at the correct answer consistently.
Real-World Applications
Understanding how to calculate areas and dimensions is not just a theoretical exercise. It has practical applications in many real-world scenarios, such as:
- Home improvement: Calculating the amount of paint or flooring needed for a room.
- Gardening: Determining the area of a garden bed or the amount of fencing required.
- Construction: Estimating the materials needed for building projects.
- Design: Laying out rooms and furniture.
By mastering these basic geometric and algebraic concepts, you'll be better equipped to tackle a wide range of practical problems.
Exploring Further
If you're interested in learning more about area calculations and geometric problems, there are many resources available online and in libraries. You can explore topics such as:
- Different geometric shapes: Learn about the area formulas for triangles, circles, and other shapes.
- Volume calculations: Extend your knowledge to three-dimensional objects.
- Trigonometry: Explore the relationships between angles and sides of triangles.
By continuing to learn and explore, you can deepen your understanding of mathematics and its applications.
Conclusion
Calculating the area of a framed portrait is a great example of how math can be applied to everyday situations. By breaking down the problem into smaller steps, understanding the relationships between dimensions, and carefully performing the calculations, you can arrive at the correct answer. Remember to avoid common mistakes and always double-check your work. With practice, you'll become more confident in your ability to solve these types of problems. Whether you're framing a portrait, planning a garden, or designing a building, the skills you've learned here will be valuable assets.
