answer:First, let's find out how many seats are currently taken. Since the auditorium holds 500 people and two-fifths of the seats are taken, we can calculate this as follows: 2/5 * 500 = 200 So, 200 seats are currently taken. Next, let's find out how many seats are broken. Since 1/10 of the seats are broken, we can calculate this as follows: 1/10 * 500 = 50 So, 50 seats are broken. To find out how many seats are still available, we need to subtract the number of seats that are taken and the number of seats that are broken from the total number of seats in the auditorium: 500 (total seats) - 200 (taken seats) - 50 (broken seats) = 250 Therefore, there are boxed{250} seats still available in the auditorium.

answer:First, let's convert the length of the spool of string from feet to inches to make it easier to work with the wick lengths, which are given in inches. There are 12 inches in a foot, so a 15-foot spool of string is: 15 feet * 12 inches/foot = 180 inches Now, let's denote the number of 6-inch wicks as x and the number of 12-inch wicks as y. We know that the total length of the wicks must add up to 180 inches, so we can write the equation: 6x + 12y = 180 Since we want an equal number of 6-inch and 12-inch wicks, we can set x = y. Let's substitute y for x in the equation: 6x + 12x = 180 Combine like terms: 18x = 180 Now, divide both sides by 18 to solve for x: x = 180 / 18 x = 10 Since x = y, we have 10 of each type of wick. Therefore, the total number of wicks is: 10 (6-inch wicks) + 10 (12-inch wicks) = 20 wicks Amy will have cut a total of boxed{20} wicks.

answer:Initially, there were 30 blonde-haired girls in the choir. After adding 10 more blonde-haired girls, the total number of blonde-haired girls becomes 30 + 10 = 40. The total number of girls in the choir before adding the new girls was 80. Since we now know there are 40 blonde-haired girls, we can find the number of black-haired girls by subtracting the number of blonde-haired girls from the total number of girls. So, the number of black-haired girls is 80 - 30 = 50. Therefore, there are boxed{50} black-haired girls in the choir.

answer:**Analysis:** This problem tests our understanding of the area and arc length formulas for sectors in radian measure. Pay attention to the distinction and connection between radian and degree measures. According to the given information, we can set up the arc length and radius of the sector. By utilizing the sector's perimeter and area, we can find the arc length and radius, and then apply the formula α = frac{l}{r} to find the central angle in radians. **Step-by-step solution:** Let the arc length of the sector be denoted as l, and the radius as r. Thus, we have: 1. 2r + l = 4 (perimeter) 2. S_{Area} = frac{1}{2}lr = 1 (area) Solving this system of equations, we obtain: r = 1 and l = 2. Therefore, the central angle of the sector in radians is: α = frac{l}{r} = frac{2}{1} = boxed{2}.