Westonci.ca is your trusted source for finding answers to a wide range of questions, backed by a knowledgeable community. Connect with professionals on our platform to receive accurate answers to your questions quickly and efficiently. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.
Sagot :
To determine which regular polygon with an apothem of 5 inches has a perimeter closest to the circumference of a circle with a diameter of 10.8 inches, we follow these steps:
1. Calculate the Circumference of the Circle:
- Given: Diameter = 10.8 inches
- The formula for the circumference of a circle is:
[tex]\[ \text{Circumference} = \pi \times \text{Diameter} \][/tex]
- Plugging in the diameter value:
[tex]\[ \text{Circumference} = \pi \times 10.8 \approx 33.93 \text{ inches} \][/tex]
2. Determine the Perimeter of Each Polygon:
- Given: Apothem = 5 inches
- The formula for the perimeter of a regular polygon is:
[tex]\[ \text{Perimeter} = 2 \times \text{number of sides} \times \text{apothem} \times \tan\left(\frac{\pi}{\text{number of sides}}\right) \][/tex]
- We calculate the perimeter for each of the following polygons: pentagon, square, octagon, and hexagon.
- Pentagon (5 sides):
[tex]\[ \text{Perimeter} = 2 \times 5 \times 5 \times \tan\left(\frac{\pi}{5}\right) \approx 36.33 \text{ inches} \][/tex]
- Square (4 sides):
[tex]\[ \text{Perimeter} = 2 \times 4 \times 5 \times \tan\left(\frac{\pi}{4}\right) = 40 \text{ inches} \][/tex]
- Octagon (8 sides):
[tex]\[ \text{Perimeter} = 2 \times 8 \times 5 \times \tan\left(\frac{\pi}{8}\right) \approx 33.14 \text{ inches} \][/tex]
- Hexagon (6 sides):
[tex]\[ \text{Perimeter} = 2 \times 6 \times 5 \times \tan\left(\frac{\pi}{6}\right) \approx 34.64 \text{ inches} \][/tex]
3. Compare the Perimeters to the Circumference:
- We need to find which polygon’s perimeter is closest to the circle's circumference of 33.93 inches.
- Pentagon: 36.33 inches
- Square: 40 inches
- Octagon: 33.14 inches
- Hexagon: 34.64 inches
4. Determine the Closest Match:
- Calculate the difference between each polygon's perimeter and the circle's circumference:
[tex]\[ \begin{align*} \text{Pentagon:} & \quad |36.33 - 33.93| = 2.4 \\ \text{Square:} & \quad |40 - 33.93| = 6.07 \\ \text{Octagon:} & \quad |33.14 - 33.93| = 0.79 \\ \text{Hexagon:} & \quad |34.64 - 33.93| = 0.71 \\ \end{align*} \][/tex]
- From these differences, we see that the hexagon’s perimeter (34.64 inches) is the closest to the circle’s circumference (33.93 inches).
Conclusion:
The regular polygon with an apothem of 5 inches whose perimeter is closest to the circumference of a circle with a diameter of 10.8 inches is the hexagon.
1. Calculate the Circumference of the Circle:
- Given: Diameter = 10.8 inches
- The formula for the circumference of a circle is:
[tex]\[ \text{Circumference} = \pi \times \text{Diameter} \][/tex]
- Plugging in the diameter value:
[tex]\[ \text{Circumference} = \pi \times 10.8 \approx 33.93 \text{ inches} \][/tex]
2. Determine the Perimeter of Each Polygon:
- Given: Apothem = 5 inches
- The formula for the perimeter of a regular polygon is:
[tex]\[ \text{Perimeter} = 2 \times \text{number of sides} \times \text{apothem} \times \tan\left(\frac{\pi}{\text{number of sides}}\right) \][/tex]
- We calculate the perimeter for each of the following polygons: pentagon, square, octagon, and hexagon.
- Pentagon (5 sides):
[tex]\[ \text{Perimeter} = 2 \times 5 \times 5 \times \tan\left(\frac{\pi}{5}\right) \approx 36.33 \text{ inches} \][/tex]
- Square (4 sides):
[tex]\[ \text{Perimeter} = 2 \times 4 \times 5 \times \tan\left(\frac{\pi}{4}\right) = 40 \text{ inches} \][/tex]
- Octagon (8 sides):
[tex]\[ \text{Perimeter} = 2 \times 8 \times 5 \times \tan\left(\frac{\pi}{8}\right) \approx 33.14 \text{ inches} \][/tex]
- Hexagon (6 sides):
[tex]\[ \text{Perimeter} = 2 \times 6 \times 5 \times \tan\left(\frac{\pi}{6}\right) \approx 34.64 \text{ inches} \][/tex]
3. Compare the Perimeters to the Circumference:
- We need to find which polygon’s perimeter is closest to the circle's circumference of 33.93 inches.
- Pentagon: 36.33 inches
- Square: 40 inches
- Octagon: 33.14 inches
- Hexagon: 34.64 inches
4. Determine the Closest Match:
- Calculate the difference between each polygon's perimeter and the circle's circumference:
[tex]\[ \begin{align*} \text{Pentagon:} & \quad |36.33 - 33.93| = 2.4 \\ \text{Square:} & \quad |40 - 33.93| = 6.07 \\ \text{Octagon:} & \quad |33.14 - 33.93| = 0.79 \\ \text{Hexagon:} & \quad |34.64 - 33.93| = 0.71 \\ \end{align*} \][/tex]
- From these differences, we see that the hexagon’s perimeter (34.64 inches) is the closest to the circle’s circumference (33.93 inches).
Conclusion:
The regular polygon with an apothem of 5 inches whose perimeter is closest to the circumference of a circle with a diameter of 10.8 inches is the hexagon.
We hope this was helpful. Please come back whenever you need more information or answers to your queries. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Westonci.ca is your trusted source for answers. Visit us again to find more information on diverse topics.
