To use the difference of two squares to find the area of Jack's rectangular garden, we can first express the area using the formula for the area of a rectangle, which is:
\[ \text{Area} = \text{length} \times \text{width} \]
In this case, the length is 30 feet and the width is 20 feet, so the area of Jack's garden is:
\[ \text{Area} = 30 \text{ ft} \times 20 \text{ ft} = 600 \text{ square feet} \]
Now, to relate this to the difference of two squares, we can think of the dimensions in a way that allows us to factor them appropriately. The difference of two squares states that:
\[ a^2 - b^2 = (a - b)(a + b) \]
If we want to show how the area can be represented in terms of the difference of two squares, we can rewrite the dimensions of the rectangle. For example, we can use the method where we treat them as differences from the square of a larger dimension:
Let's consider the two dimensions:
- \( a = 25 \) (which is the average of 20 and 30)
- \( b = 5 \) (which is half the difference between 20 and 30)
Hence, we can represent the area as:
\[ \text{Area} = (25 + 5)(25 - 5) \]
This becomes:
\[ \text{Area} = 30 \times 20 = (25 + 5)(25 - 5) = 25^2 - 5^2 = 625 - 25 = 600 \]
Thus, Jack can use the expression \( 25^2 - 5^2 \) to represent the area of his garden using the difference of two squares.
So the answer is:
- Area = \( 25^2 - 5^2 \)
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