Question
John is putting a fence around his garden that is shaped like a half circle and a rectangle.
A rectangle has a length of 14 feet and width of 7 feet. A semicircle with diameter of 7 feet is on top of the rectangle.
How much fencing will John need? Use StartFraction 22 over 7 EndFraction for Pi.
32 ft
39 ft
46 ft
57 ft
A rectangle has a length of 14 feet and width of 7 feet. A semicircle with diameter of 7 feet is on top of the rectangle.
How much fencing will John need? Use StartFraction 22 over 7 EndFraction for Pi.
32 ft
39 ft
46 ft
57 ft
Answers
GPT-4o mini
To calculate the total amount of fencing needed for John’s garden, we need to find the perimeter of both the rectangle and the semicircle.
1. **Calculate the perimeter of the rectangle.**
The rectangle has a length of 14 feet and a width of 7 feet. However, when fencing around the garden, we will only need to account for the sides of the rectangle that are not covered by the semicircle. Therefore, we will consider:
- One length (14 ft)
- One width (7 ft)
- The other width (7 ft) is not counted because it is where the semicircle is placed.
The perimeter contribution from the rectangle is:
\[
\text{Perimeter of the rectangle} = 14 + 7 = 21 \text{ feet}
\]
2. **Calculate the circumference of the semicircle.**
The semicircle has a diameter of 7 feet. The radius \(r\) is half of the diameter:
\[
r = \frac{7}{2} = 3.5 \text{ feet}
\]
The circumference of a full circle is given by \(C = 2\pi r\). Therefore, the circumference of the semicircle is half of that:
\[
\text{Circumference of the semicircle} = \frac{1}{2} \times 2\pi r = \pi r
\]
Substituting values, using \(\frac{22}{7}\) for \(\pi\):
\[
\text{Circumference of the semicircle} = \frac{22}{7} \times 3.5
\]
First, convert 3.5 into a fraction:
\[
3.5 = \frac{7}{2}
\]
So the semicircle's circumference becomes:
\[
\text{Circumference of the semicircle} = \frac{22}{7} \times \frac{7}{2} = \frac{22 \times 7}{7 \times 2} = \frac{22}{2} = 11 \text{ feet}
\]
3. **Total fencing required.**
Now we add the fencing for the rectangle and the semicircle:
\[
\text{Total fencing} = \text{Perimeter of the rectangle} + \text{Circumference of the semicircle}
\]
\[
\text{Total fencing} = 21 + 11 = 32 \text{ feet}
\]
Therefore, John will need **32 feet** of fencing.
The correct answer is:
\[
\boxed{32 \text{ ft}}
\]
1. **Calculate the perimeter of the rectangle.**
The rectangle has a length of 14 feet and a width of 7 feet. However, when fencing around the garden, we will only need to account for the sides of the rectangle that are not covered by the semicircle. Therefore, we will consider:
- One length (14 ft)
- One width (7 ft)
- The other width (7 ft) is not counted because it is where the semicircle is placed.
The perimeter contribution from the rectangle is:
\[
\text{Perimeter of the rectangle} = 14 + 7 = 21 \text{ feet}
\]
2. **Calculate the circumference of the semicircle.**
The semicircle has a diameter of 7 feet. The radius \(r\) is half of the diameter:
\[
r = \frac{7}{2} = 3.5 \text{ feet}
\]
The circumference of a full circle is given by \(C = 2\pi r\). Therefore, the circumference of the semicircle is half of that:
\[
\text{Circumference of the semicircle} = \frac{1}{2} \times 2\pi r = \pi r
\]
Substituting values, using \(\frac{22}{7}\) for \(\pi\):
\[
\text{Circumference of the semicircle} = \frac{22}{7} \times 3.5
\]
First, convert 3.5 into a fraction:
\[
3.5 = \frac{7}{2}
\]
So the semicircle's circumference becomes:
\[
\text{Circumference of the semicircle} = \frac{22}{7} \times \frac{7}{2} = \frac{22 \times 7}{7 \times 2} = \frac{22}{2} = 11 \text{ feet}
\]
3. **Total fencing required.**
Now we add the fencing for the rectangle and the semicircle:
\[
\text{Total fencing} = \text{Perimeter of the rectangle} + \text{Circumference of the semicircle}
\]
\[
\text{Total fencing} = 21 + 11 = 32 \text{ feet}
\]
Therefore, John will need **32 feet** of fencing.
The correct answer is:
\[
\boxed{32 \text{ ft}}
\]