Final answer:
To find the amount of fencing needed to surround the circular sector, you need to calculate the circumference of the sector. The answer is approximately 856.65 feet of fencing needed. To find the area that needs to be paved, you need to calculate the area of the circular sector. The answer is approximately 7305 square feet that need to be paved (rounded to the nearest foot).
Step-by-step explanation:
To find the amount of fencing needed to surround the circular sector, you need to calculate the circumference of the sector. The formula for the circumference of a circle is C = 2πr. The central angle of the sector is 132º, which is a third of a full circle (360º). Therefore, the sector's central angle is also a third of the circumference of the full circle. The circumference of the full circle can be calculated as 2πr, where r is the radius. Given that the straight side of the garden measures 50 feet, we can find the radius by dividing the straight side length by the central angle in degrees.
radius = straight side length / (central angle / 360º)
radius = 50 feet / (132º / 360º)
radius = 50 feet / 0.3667
radius ≈ 136.29 feet
Now that we have the radius, we can calculate the circumference of the full circle:
C = 2πr
C = 2π(136.29 feet)
C ≈ 856.65 feet
Therefore, approximately 856.65 feet of fencing is needed to surround the circular sector.
b) To find the area that needs to be paved, you need to calculate the area of the circular sector. The formula for the area of a circular sector is A = (θ/360º)πr², where θ is the central angle in degrees and r is the radius. Given that the central angle is 132º, the radius is 136.29 feet (as calculated earlier).
A = (132º/360º)π(136.29 feet)² = (0.3667)π(18586.48 feet²) ≈ 7304.91 square feet (rounded to the nearest foot)
Therefore, approximately 7305 square feet of the garden needs to be paved.