To use the difference of two squares to find the area of Jack's rectangular garden, we need to express the area in a form that matches the difference of squares formula, which is \( a^2 - b^2 = (a - b)(a + b) \).
The area of Jack's garden is given by the formula for the area of a rectangle:
\[ \text{Area} = \text{length} \times \text{width} = 30 , \text{feet} \times 20 , \text{feet} = 600 , \text{square feet}. \]
We look for an expression among the options provided that can simplify or equate to 600.
Now let's evaluate the choices:
a) \( 25^2 - 5^2 \)
Calculating this:
\[ 25^2 = 625, \quad 5^2 = 25 \quad \Rightarrow \quad 25^2 - 5^2 = 625 - 25 = 600 \quad \text{(this works)}. \]
b) \( 20^2 - 10^2 \)
Calculating this:
\[ 20^2 = 400, \quad 10^2 = 100 \quad \Rightarrow \quad 20^2 - 10^2 = 400 - 100 = 300 \quad \text{(does not work)}. \]
c) \( 15^2 - 10^2 \)
Calculating this:
\[ 15^2 = 225, \quad 10^2 = 100 \quad \Rightarrow \quad 15^2 - 10^2 = 225 - 100 = 125 \quad \text{(does not work)}. \]
d) \( 35^2 - 15^2 \)
Calculating this:
\[ 35^2 = 1225, \quad 15^2 = 225 \quad \Rightarrow \quad 35^2 - 15^2 = 1225 - 225 = 1000 \quad \text{(does not work)}. \]
The only option that gives us the area of Jack's garden, which is 600 square feet, is:
a) \( 25^2 - 5^2 \).
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