To solve the problem of finding the dimensions of the garden and its area, we need to take into account the total fencing available and the width of the border. Let's walk through the solution step by step.
1. Understanding the Total Fencing:
The gardener has 97 feet of fencing to enclose a rectangular garden that has a 2-feet wide border surrounding it. Given that the length and width of the garden are the same, we know it's a square garden.
2. Calculating the Outer Perimeter:
Since the garden is square and has a border, the total length of fencing available (97 feet) must enclose the outer perimeter, including the border. Let's denote the side length of the square garden as [tex]\( s \)[/tex]. The outer side length, including the 2-feet border on each side, is [tex]\( s + 4 \)[/tex] feet.
3. Forming the Equation:
The perimeter of the outer square (including the border) is:
[tex]\[
4 \times (s + 4) = 97
\][/tex]
4. Solving for [tex]\( s \)[/tex]:
[tex]\[
s + 4 = \frac{97}{4}
\][/tex]
[tex]\[
s + 4 = 24.25
\][/tex]
[tex]\[
s = 24.25 - 4
\][/tex]
[tex]\[
s = 20.25
\][/tex]
5. Calculating the Area of the Garden:
The side length of the garden without the border is 20.2 feet (rounded to the nearest tenth). To find the area of the square garden:
[tex]\[
\text{Area} = (s)^2 = (20.2)^2 = 20.2 \times 20.2 = 410.0
\][/tex]
Notice in the final rounded form:
[tex]\[
\text{Area} = 410.1 \text{ square feet}
\][/tex]
Thus, the dimensions of the garden are 20.2 feet by 20.2 feet, and the area is 410.1 square feet.
