To determine the maximum length of the diameter \( d \) for the semicircular garden using 33 feet of fencing, we start with the formula for the circumference \( C \) of a full circle, which is given by:
\[ C = \pi \times d \]
Since we are only using a semicircle, the fence will cover half of the circumference. Thus, the fencing needed for the semicircle can be expressed as:
\[ \text{Fencing length} = \frac{1}{2} \pi d \]
Given that we have 33 feet of fencing, we can set up the following inequality:
\[ \frac{1}{2} \pi d \leq 33 \]
To simplify this inequality, we can multiply both sides by 2:
\[ \pi d \leq 66 \]
Next, we can solve for \( d \) by dividing by \( \pi \). Using \( \pi \approx 227/100 \) for our calculations, we rewrite the inequality:
\[ d \leq \frac{66}{\pi} \]
Substituting in the approximate value of \( \pi \):
\[ d \leq \frac{66}{227/100} \]
This simplifies to:
\[ d \leq \frac{66 \times 100}{227} \approx \frac{6600}{227} \approx 29.05 \]
However, we can work with the more approximate value of \( \pi \) itself. Hence, let’s use the provided choices to express the solution of the inequality for the garden diameter \( d \).
Looking at your options:
- \( d \leq 10.5 \)
- \( d \leq 21 \)
- \( d \leq 42 \)
- \( d \leq 104 \)
We can compare our derived value, which indicates that the maximum diameter we establish and bound against \( 33 \) feet is actually less than \( 42 \) feet but greater than any of the options below \( 21 \).
Thus, to finalize the solution, we conclude with:
The solution of the inequality is \( d \leq 21 \) (Option 3).
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