answer:To solve this, we use the fact that the common chord of the two circles always passes through the center of the circle (x+1)^{2}+(y+1)^{2}=4. Convert both circles into the general form of the equation, and by subtracting them, we can obtain the equation of the common chord as (2a+2)x+(2b+2)y-a^{2}-1=0. This line passes through the center of the circle (-1,-1). Substituting this into the equation, we get a^{2}+2a+2b+5=0. Therefore, the correct answer is boxed{text{B: } a^{2}+2a+2b+5=0}.

answer:To calculate the total earnings of the amusement park for the week, we can break down the calculation into the number of visitors each day and then multiply by the ticket price. 1. During the weekdays (Monday to Friday), the park was visited by 100 people each day. Since there are 5 weekdays, the total number of visitors during the weekdays is: [100 text{ people/day} times 5 text{ days} = 500 text{ people}] 2. On Saturday, the park was visited by 200 people, and on Sunday, it was visited by 300 people. So, the total number of visitors during the weekend is: [200 text{ people (Saturday)} + 300 text{ people (Sunday)} = 500 text{ people}] 3. Adding the visitors from the weekdays and the weekend gives us the total number of visitors for the week: [500 text{ people (weekdays)} + 500 text{ people (weekend)} = 1000 text{ people}] 4. Since each ticket is sold for 3, the total earnings for the week can be calculated by multiplying the total number of visitors by the ticket price: [1000 text{ people} times 3/text{person} = 3000] Therefore, the amusement park made a total of boxed{3000} in a week.

answer:To find the average speed of the car, we need to calculate the total distance traveled and the total time taken. The total distance traveled is the sum of the uphill and downhill distances: Total distance = 100 km (uphill) + 50 km (downhill) = 150 km Now, we need to find the time taken for each part of the trip. Time taken to travel uphill: Speed = 30 km/hr Distance = 100 km Time = Distance / Speed = 100 km / 30 km/hr = 10/3 hours Time taken to travel downhill: Speed = 70 km/hr Distance = 50 km Time = Distance / Speed = 50 km / 70 km/hr = 5/7 hours Total time taken is the sum of the time taken uphill and downhill: Total time = (10/3) hours + (5/7) hours To add these two times, we need a common denominator, which is 21 (3 * 7): Total time = (10/3) * (7/7) + (5/7) * (3/3) Total time = (70/21) + (15/21) Total time = (70 + 15) / 21 Total time = 85/21 hours Now, we can calculate the average speed: Average speed = Total distance / Total time Average speed = 150 km / (85/21) hours To divide by a fraction, we multiply by its reciprocal: Average speed = 150 km * (21/85) hours^-1 Average speed = (150 * 21) / 85 km/hr Average speed = 3150 / 85 km/hr Average speed ≈ 37.06 km/hr So, the average speed of the car is approximately boxed{37.06} km/hr.

answer:Let's start by calculating the total number of legs for the people in the pool and those not in the pool. We know that humans have 2 legs each. So, if there are 16 legs in the pool, we can divide that number by 2 to find out how many people are in the pool: 16 legs ÷ 2 legs/person = 8 people in the pool Now, let's add the number of people in the pool to the number of people not in the pool to find the total number of people: 8 people in the pool + 6 people not in the pool = 14 people total We know that Karen and Donald have 6 children, so that's 8 people in their family (Karen, Donald, and their 6 children). We also know that Tom and Eva are part of the group, so that's 2 more people: 8 (Karen and Donald's family) + 2 (Tom and Eva) = 10 people Now, we subtract the number of people we know (Karen, Donald, Tom, Eva, and Karen and Donald's 6 children) from the total number of people to find out how many children Tom and Eva have: 14 people total - 10 people (Karen, Donald, Tom, Eva, and Karen and Donald's children) = 4 children Therefore, Tom and Eva have boxed{4} children.