Let's solve the problem step by step:
### Step 1: Understand the problem
We have a square garden plot with an area of 75 square feet, and we need to determine:
a. The length of each side in simplest radical form.
b. The length of each side to the nearest tenth of a foot.
### Step 2: Setup and solve for the side length
Definition: The area [tex]\( A \)[/tex] of a square with side length [tex]\( s \)[/tex] is given by:
[tex]\[ A = s^2 \][/tex]
Given that [tex]\( A = 75 \)[/tex] square feet, we need to solve for [tex]\( s \)[/tex].
### Step 3: Find the side length in simplest radical form
To find the side length [tex]\( s \)[/tex] from the area:
[tex]\[ s = \sqrt{A} \][/tex]
[tex]\[ s = \sqrt{75} \][/tex]
Next, simplify [tex]\( \sqrt{75} \)[/tex]:
[tex]\[ 75 = 25 \times 3 \][/tex]
[tex]\[ \sqrt{75} = \sqrt{25 \times 3} \][/tex]
[tex]\[ \sqrt{75} = \sqrt{25} \times \sqrt{3} \][/tex]
[tex]\[ \sqrt{75} = 5 \sqrt{3} \][/tex]
So, the simplest radical form for the side length is [tex]\( 5 \sqrt{3} \)[/tex].
### Step 4: Find the side length to the nearest tenth of a foot
To find the decimal form of [tex]\( 5 \sqrt{3} \)[/tex]:
[tex]\[ 5 \sqrt{3} \approx 5 \times 1.732 \approx 8.66 \][/tex]
Rounding 8.66 to the nearest tenth, we get:
[tex]\[ 5 \sqrt{3} \approx 8.7 \][/tex]
### Step 5: Match the results with the given options
The results we have found are:
a. The simplest radical form: [tex]\( 5 \sqrt{3} \)[/tex]
b. The length to the nearest tenth: [tex]\( 8.7 \)[/tex] feet
Looking at the given options:
1. [tex]\( \frac{75}{4} ; 18.8 ft \)[/tex]
2. [tex]\( \sqrt{75} ; 9 ft \)[/tex]
3. [tex]\( \frac{\sqrt{75}}{2} ; 4.33 ft \)[/tex]
4. [tex]\( 5 \sqrt{3} ; 8.7 ft \)[/tex]
The correct pair that matches our findings is:
[tex]\[ 5 \sqrt{3} ; 8.7 ft \][/tex]
### Conclusion
Therefore, the correct answer to the problem is:
[tex]\[ 5 \sqrt{3} ; 8.7 ft \][/tex]
