answer:Given the radius of the arc is 24 and the central angle it subtends is 60^{circ}, we can find the length of the arc using the formula l = frac{theta}{360} times 2pi r, where theta is the central angle in degrees and r is the radius of the arc. Starting with the given values, we have: [ l = frac{60}{360} times 2pi times 24 ] Simplifying the fraction frac{60}{360} to frac{1}{6}, we get: [ l = frac{1}{6} times 2pi times 24 ] Multiplying through, we find: [ l = frac{2pi times 24}{6} = frac{48pi}{6} = 8pi ] Therefore, the length of the arc is boxed{8pi}.

answer:Since G is the centroid of triangle ABC, it divides medians overline{AM} and overline{BN} in the ratio 2:1. Thus, AG = frac{2}{3} cdot AM = frac{2}{3} cdot 15 = 10 and BG = frac{2}{3} cdot BN = frac{2}{3} cdot 20 = frac{40}{3}. Applying the Pythagorean Theorem to triangle ABG (since AG perp BG): [ AG^2 + BG^2 = AB^2 ] [ 10^2 + left(frac{40}{3}right)^2 = AB^2 ] [ 100 + frac{1600}{9} = AB^2 ] [ frac{900 + 1600}{9} = AB^2 ] [ frac{2500}{9} = AB^2 ] [ AB = frac{50}{3} ] Since overline{BN} is median to side AC, AC = 2AB = 2 times frac{50}{3} = frac{100}{3}. Conclusion: The length of side AC in triangle ABC is boxed{frac{100}{3}}.

answer:Let's denote the side length of the large pizza as ( x ) inches. The area of the small pizza is ( 6 times 6 = 36 ) square inches. The area of the large pizza is ( x times x = x^2 ) square inches. The cost of the small pizza is 10, and the cost of the large pizza is 20. If each friend spends 30 alone, they can buy: - Friend 1: ( frac{30}{10} times 36 = 3 times 36 = 108 ) square inches of small pizza. - Friend 2: ( frac{30}{20} times x^2 = 1.5 times x^2 ) square inches of large pizza. If they pool their money, they have 60 to spend. They can buy: - ( frac{60}{20} times x^2 = 3 times x^2 ) square inches of large pizza. According to the problem, pooling their money gives them 9 more square inches of pizza than buying separately. So we have: [ 3 times x^2 = 108 + 1.5 times x^2 + 9 ] Simplify the equation: [ 3x^2 = 117 + 1.5x^2 ] Subtract ( 1.5x^2 ) from both sides: [ 3x^2 - 1.5x^2 = 117 ] [ 1.5x^2 = 117 ] Divide both sides by 1.5: [ x^2 = frac{117}{1.5} ] [ x^2 = 78 ] Take the square root of both sides: [ x = sqrt{78} ] [ x approx 8.83 ] So the side length of the large pizza is approximately boxed{8.83} inches.

answer:To find out how many years Joanie is paying for, we first need to subtract the down payment from the total cost to find out how much she is paying in monthly fees. Total cost = 482 Down payment = 50 Total cost - Down payment = Monthly fees 482 - 50 = 432 Now we know that Joanie will be paying 432 in monthly fees. Since the gym costs 12 per month, we divide the total monthly fees by the monthly cost to find out how many months she is paying for. Monthly fees = 432 Monthly cost = 12 Number of months = Monthly fees / Monthly cost Number of months = 432 / 12 Number of months = 36 Now we know that Joanie is paying for 36 months. Since there are 12 months in a year, we divide the number of months by 12 to find out how many years she is paying for. Number of months = 36 Months in a year = 12 Number of years = Number of months / Months in a year Number of years = 36 / 12 Number of years = 3 Joanie is paying for boxed{3} years.