- Define the length \(L\) and width \(W\) as algebraic expressions.
- Calculate the difference \(D = L - W\) by subtracting the expressions.
- Distribute the negative sign and combine like terms to simplify the expression.
- Identify the correct option as \(\boxed{9x + 4x^2 - 5y}\).
Explanation
- Analysis of the problem
To find how much greater the length of the field is than the width, we need to calculate the difference between the given expressions for length \(L\) and width \(W\).
The length is given as \(L = 14x - 3x^2 + 2y\).
The width is given as \(W = 5x - 7x^2 + 7y\).
Our objective is to evaluate \(D = L - W\).
- Setting up the subtraction
We set up the subtraction:
\[D = (14x - 3x^2 + 2y) - (5x - 7x^2 + 7y)\]
To perform the subtraction, we distribute the negative sign to each term inside the second set of parentheses:
\[D = 14x - 3x^2 + 2y - 5x + 7x^2 - 7y\]
- Simplifying and combining like terms
Now, we group the like terms together:
\[D = (-3x^2 + 7x^2) + (14x - 5x) + (2y - 7y)\]
Next, we combine the coefficients for each variable:
- For the \(x^2\) term: \(-3 + 7 = 4\), so we have \(4x^2\).
- For the \(x\) term: \(14 - 5 = 9\), so we have \(9x\).
- For the \(y\) term: \(2 - 7 = -5\), so we have \(-5y\).
Combining these, we get:
\[D = 4x^2 + 9x - 5y\]
Rearranging to match the format of the given options:
\[D = 9x + 4x^2 - 5y\]
- Conclusion
Comparing our result \(9x + 4x^2 - 5y\) with the provided options, it matches the first choice. Thus, the length is greater than the width by \(9x + 4x^2 - 5y\).
The final answer is \(\boxed{9x + 4x^2 - 5y}\).
Examples
In landscape architecture or construction, you often need to compare the dimensions of different garden beds or plots to ensure they meet design specifications. For example, if you are designing a rectangular lawn and a flower bed, you might represent their lengths and widths as algebraic expressions. Calculating the difference between these expressions allows you to determine exactly how much extra material, such as border edging or turf, you need for the larger section, ensuring precision in your planning and budgeting.