Question

Jack has a rectangular garden measuring 20 feet by 30 feet. How can he use the difference of two squares to find the area of the garden? Select from the following expressions:
a) 25^2 -5^2
b)20^2 -10^2
c)15^2 -10^2
d)35^2 -15^2

Answer 1

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) \( 25^2 - 5^2 \)**.