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Sagot :
To solve the problem of calculating the amount of fencing Jordana will need, we need to look at the combined perimeter of the garden's half-circle and rectangular parts.
1. Understand the Dimensions:
- The rectangular part has a length of 64 feet and a width of 32 feet (64 divided by 2).
- The semicircle's diameter equals the rectangle's width, which is 32 feet. Therefore, the radius of the semicircle is 16 feet (32 divided by 2).
2. Perimeter of the Rectangular Part:
- To get the perimeter of the rectangular part:
- The rectangle's perimeter formula is [tex]\( \text{Perimeter} = 2 \times \text{Length} + 2 \times \text{Width} \)[/tex].
- Since the semicircle covers one width of the rectangle, we modify the perimeter formula by subtracting one width:
- [tex]\[ \text{Perimeter of rectangle} = 2 \times 64 + 2 \times 32 - 32 \\ = 2 \times 64 + 32 \\ = 128 + 32 = 160 \text{ feet} \][/tex]
3. Circumference of the Semicircle:
- For a semicircle, the circumference is half the full circle's circumference plus the diameter of the semicircle (which is along the bottom edge of the rectangular part).
- The formula for the circumference of a circle is [tex]\(2\pi r\)[/tex], so the semicircle's circumference is given by:
- [tex]\[ \text{Circumference of semicircle} = \frac{\pi \times \text{diameter}}{2} \\ \text{which equals } \pi \times \text{radius} \][/tex]
- With the radius (r) being 16 feet and using given [tex]\(\pi = \frac{22}{7}\)[/tex]:
- [tex]\[ \text{Circumference of semicircle} = \frac{22}{7} \times 16 \\ = 50.285714285714285 \text{ feet} \][/tex]
4. Total Fencing Needed:
- Add together the modified rectangular perimeter and the semicircle circumference to find the total fencing required:
- [tex]\[ \text{Total fencing} = 160 + 50.285714285714285 \\ \approx 210.28571428571428 \text{ feet} \][/tex]
So, Jordana will need approximately [tex]\( 210.28571428571428 \)[/tex] feet of fencing to enclose her garden.
The closest choice to this calculation is none of the given options [tex]\(64 ft, 86 ft, 92 ft,\)[/tex] or [tex]\(114 ft\)[/tex]. Thus, it seems this question might have an incorrect multiple-choice answer set. The correct calculated amount of fencing she needs is 210.28571428571428 feet.
1. Understand the Dimensions:
- The rectangular part has a length of 64 feet and a width of 32 feet (64 divided by 2).
- The semicircle's diameter equals the rectangle's width, which is 32 feet. Therefore, the radius of the semicircle is 16 feet (32 divided by 2).
2. Perimeter of the Rectangular Part:
- To get the perimeter of the rectangular part:
- The rectangle's perimeter formula is [tex]\( \text{Perimeter} = 2 \times \text{Length} + 2 \times \text{Width} \)[/tex].
- Since the semicircle covers one width of the rectangle, we modify the perimeter formula by subtracting one width:
- [tex]\[ \text{Perimeter of rectangle} = 2 \times 64 + 2 \times 32 - 32 \\ = 2 \times 64 + 32 \\ = 128 + 32 = 160 \text{ feet} \][/tex]
3. Circumference of the Semicircle:
- For a semicircle, the circumference is half the full circle's circumference plus the diameter of the semicircle (which is along the bottom edge of the rectangular part).
- The formula for the circumference of a circle is [tex]\(2\pi r\)[/tex], so the semicircle's circumference is given by:
- [tex]\[ \text{Circumference of semicircle} = \frac{\pi \times \text{diameter}}{2} \\ \text{which equals } \pi \times \text{radius} \][/tex]
- With the radius (r) being 16 feet and using given [tex]\(\pi = \frac{22}{7}\)[/tex]:
- [tex]\[ \text{Circumference of semicircle} = \frac{22}{7} \times 16 \\ = 50.285714285714285 \text{ feet} \][/tex]
4. Total Fencing Needed:
- Add together the modified rectangular perimeter and the semicircle circumference to find the total fencing required:
- [tex]\[ \text{Total fencing} = 160 + 50.285714285714285 \\ \approx 210.28571428571428 \text{ feet} \][/tex]
So, Jordana will need approximately [tex]\( 210.28571428571428 \)[/tex] feet of fencing to enclose her garden.
The closest choice to this calculation is none of the given options [tex]\(64 ft, 86 ft, 92 ft,\)[/tex] or [tex]\(114 ft\)[/tex]. Thus, it seems this question might have an incorrect multiple-choice answer set. The correct calculated amount of fencing she needs is 210.28571428571428 feet.
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