Question

You want to build a semicircular garden by placing a semicircle of fencing against a wall. You have 33 feet of fencing you can use. What is the solution of an inequality that shows how many feet long the diameter, d , of the garden can be? Use 227 for pi. (Hint: The circumference of a circle equals pi times the diameter.)(1 point)

Answer 1

To find the solution to the inequality for the diameter \( d \) of the semicircular garden, we start by recognizing how much fencing is needed.

The perimeter (or circumference) of a semicircle is given by the formula:

\[ \text{Perimeter of semicircle} = \frac{1}{2} \pi d + d \]

Here, \(\frac{1}{2} \pi d\) represents the curved part of the semicircle, and \(d\) is the straight part (the diameter) that is against the wall.

Given that the total amount of fencing available is 33 feet, we can set up the following equation with the approximation of \(\pi\) being 227/100:

\[ \frac{1}{2} \cdot \frac{227}{100} d + d \leq 33 \]

Combining the terms related to \( d \):

\[ \frac{227}{200} d + d \leq 33 \]

To combine the terms, we express \( d \) using a common denominator:

\[ \frac{227}{200} d + \frac{200}{200} d \leq 33 \] \[ \frac{427}{200} d \leq 33 \]

Next, we solve for \( d \) by multiplying both sides of the inequality by \( \frac{200}{427} \):

\[ d \leq 33 \cdot \frac{200}{427} \]

Calculating the right side:

\[ 33 \cdot \frac{200}{427} \approx \frac{6600}{427} \approx 15.43 \]

Thus, the solution to the inequality showing how many feet long the diameter \( d \) of the garden can be is:

\[ d \leq 15.43 \]

This means the diameter of the garden must be less than or equal to approximately 15.43 feet.