How to Solve a Word Problem Involving Area and Borders
Quick Answer
To find the width of the border around a framed photo, set up the equation based on the total area. The outer dimensions can be expressed as (10 + 2x) and (6 + 2x), leading to the equation (10 + 2x)(6 + 2x) = 96.
Understanding how to solve word problems that involve area, such as the one about a framed photo with a border, is an important skill in math. In this problem, you need to find the width of the border surrounding a picture that measures 10 inches by 6 inches and has a total area of 96 square inches when framed.
To start, let's define the width of the border as \(x\) inches. This means that the border contributes to both the length and the width of the overall framed picture. The new dimensions of the framed picture will be:
- Outer length = original length + border on both sides = \(10 + 2x\)
- Outer width = original width + border on both sides = \(6 + 2x\)
Next, we know that the area of a rectangle is calculated by multiplying its length by its width. Therefore, the area of the framed picture can be expressed by the equation:
\[(10 + 2x)(6 + 2x) = 96\]
Now, let's expand the left side of the equation. Using the distributive property (also known as the FOIL method for binomials), we multiply:
\[(10 + 2x)(6 + 2x) = 10 \cdot 6 + 10 \cdot 2x + 2x \cdot 6 + 2x \cdot 2x\]
This simplifies to:
\[60 + 20x + 12x + 4x^2\]
Combining like terms, we have:
\[4x^2 + 32x + 60 = 96\]
To solve for \(x\), we want to set the equation to zero. We can do this by subtracting 96 from both sides:
\[4x^2 + 32x + 60 - 96 = 0\]
This simplifies to:
\[4x^2 + 32x - 36 = 0\]
Now, we can simplify this equation further by dividing everything by 4:
\[x^2 + 8x - 9 = 0\]
Next, we can factor this quadratic equation. Look for two numbers that multiply to \(-9\) and add to \(8\). The numbers \(9\) and \(-1\) work:
\[(x + 9)(x - 1) = 0\]
Setting each factor to zero gives us:
\[x + 9 = 0 \Rightarrow x = -9\] (not a valid solution since width cannot be negative)
\[x - 1 = 0 \Rightarrow x = 1\] (valid solution)
Thus, the width of the border is 1 inch.
This kind of problem is not just theoretical; it has real-world applications. For instance, understanding how to determine dimensions and areas can help in home decoration, designing picture frames, or even landscaping. By practicing problems like this, you can improve your problem-solving skills and gain confidence in your math abilities.
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