Sure, let's solve the problem step by step:
### Part (a): Find the length of each side in simplest radical form.
1. Understand the problem: We are given the area of a square garden plot, which is [tex]\( 24 \text{ ft}^2 \)[/tex]. We need to determine the length of each side in simplest radical form.
2. Formula: The area of a square is given by [tex]\( \text{Area} = \text{side}^2 \)[/tex].
3. Set up the equation: Let [tex]\( s \)[/tex] be the length of a side of the square. We have:
[tex]\[
s^2 = 24
\][/tex]
4. Solve for [tex]\( s \)[/tex]: Take the square root of both sides:
[tex]\[
s = \sqrt{24}
\][/tex]
5. Simplify the radical: To express [tex]\( \sqrt{24} \)[/tex] in simplest radical form, note that:
[tex]\[
24 = 4 \times 6 \implies \sqrt{24} = \sqrt{4 \times 6} = \sqrt{4} \times \sqrt{6} = 2\sqrt{6}
\][/tex]
So, the length of each side in simplest radical form is:
[tex]\[
s = 2\sqrt{6} \text{ feet}
\][/tex]
### Part (b): Calculate the length of each side to the nearest tenth of a foot.
1. Use the simplified radical form: We found that [tex]\( s = 2\sqrt{6} \)[/tex].
2. Approximate the value numerically: Calculate [tex]\( 2\sqrt{6} \)[/tex]. Typically, the calculator or approximation of [tex]\( \sqrt{6} \approx 2.449 \)[/tex]:
[tex]\[
2 \times 2.449 = 4.898
\][/tex]
3. Round to the nearest tenth: The value [tex]\( 4.898 \)[/tex] rounded to the nearest tenth is:
[tex]\[
4.9 \text{ feet}
\][/tex]
### Summary of the Results
- The length of each side in simplest radical form is [tex]\( 2\sqrt{6} \)[/tex] feet.
- The length of each side rounded to the nearest tenth of a foot is [tex]\( 4.9 \)[/tex] feet.
Given the multiple-choice options, the correct one is:
[tex]\[ 2\sqrt{6} ; 4.9 \text{ ft} \][/tex]
