The circumference of the circle is given by C = πd, where d is the diameter of the circle. For the beverage area, d = 50 feet, so the circumference is C = π(50) = 157.08 feet (rounded to two decimal places). Each booth needs 10.5 feet (arc length) between its center and the center of the booth next to it. To determine how many booths can fit, we need to subtract the arc length between booths from the circumference of the circle and then divide by the arc length of each booth:
Number of booths = (C - nL) / L
where n is the number of spaces between the booths and L is the arc length of each booth. We can rearrange this equation to solve for n:
n = (C - L x Number of booths) / L
Substituting the values, we get:
n = (157.08 - 10.5x) / 10.5
where x is the number of booths. To find the maximum value of x, we need to round down to the nearest whole number:
x = floor(n) = floor((157.08 - 10.5x) / 10.5) = 14
Therefore, 14 booths can fit in the beverage area.
The circumference of the circle is C = πd = π(100) = 314.16 feet (rounded to two decimal places). The angle between each pole is 15 degrees, so the angle between the centers of each booth is also 15 degrees. To find the arc length between the centers of each booth, we need to multiply the circumference of the circle by the ratio of the angle between the centers of each booth to the angle between the poles:
Arc length between centers of each booth = C x (15/360) = 13.09 feet (rounded to two decimal places)
Therefore, there will be approximately 13.1 feet between the centers of each booth.
The circumference of the circle is C = πd = π(150) = 471.24 feet (rounded to two decimal places). Each food booth needs 30 feet between its center and the center of the booth next to it. To determine how many booths can fit, we can use the same equation as in part 1:
Number of booths = (C - nL) / L
where L = 30 feet. Substituting the values, we get:
x = floor((471.24 - 30x) / 30) = 15
Therefore, there will be approximately 15 food booths.
I hope this helps!