answer：First, I need to determine the total distance Benjamin walked on the last two days of his hiking trip.He walked 8 miles on the first day and 11 miles on the second day, which adds up to 19 miles for the first two days.The total distance for the entire four-day trip is 35 miles. To find out how much he walked on the last two days, I subtract the distance covered in the first two days from the total distance.So, 35 miles minus 19 miles equals 16 miles.Therefore, Benjamin walked a total of 16 miles on the last two days of his trip.

answer：First, I need to calculate the total cost of the notebooks. The library bought 30 packs, and each pack contains 8 cases costing 3 each. So, the cost per pack of notebooks is 8 multiplied by 3, which equals 24. Then, multiplying this by the number of packs, 30, gives the total cost for notebooks: 30 multiplied by 24 equals 720.Next, I'll calculate the total cost of the pens. The library purchased 15 packs, and each pack has 6 cases costing 5 each. The cost per pack of pens is 6 multiplied by 5, which equals 30. Multiplying this by the number of packs, 15, gives the total cost for pens: 15 multiplied by 30 equals 450.Finally, to find the total amount spent by the library, I'll add the total cost of notebooks and the total cost of pens together: 720 plus 450 equals 1,170.

answer：First, I'll identify the variables involved in the problem. Let ( C ) represent the time Emily spent preparing her costume.According to the problem, Emily rehearsed her dance routine three times as long as she spent preparing her costume. Therefore, the rehearsal time ( R ) can be expressed as:[ R = 3C ]Next, the problem states that she spent twice as long designing the stage set as she did rehearsing her dance routine. Let ( S ) represent the time spent designing the stage set. Thus:[ S = 2R ]We are given that Emily spent 60 hours designing the stage set, so:[ S = 60 ]Substituting the expression for ( R ) into the equation for ( S ):[ 60 = 2 times 3C ][ 60 = 6C ]Solving for ( C ):[ C = frac{60}{6} ][ C = 10 ]Therefore, Emily spent 10 hours preparing her costume.

answer：First, I need to determine the heights of each person based on the given information.Starting with Lucas, since Olivia is 55 inches tall and Lucas is 10 inches taller than Olivia, Lucas's height is 55 + 10 = 65 inches.Next, Ethan is 20 inches shorter than Lucas. So, Ethan's height is 65 - 20 = 45 inches.Finally, Mia is 3 inches taller than Ethan. Therefore, Mia's height is 45 + 3 = 48 inches.

answer：To find the value of ( b - a ), I start by expanding the expression ( (x - a)^2 - 1 ).Expanding ( (x - a)^2 ) gives ( x^2 - 2ax + a^2 ). Subtracting 1 results in ( x^2 - 2ax + a^2 - 1 ).Next, I compare this expanded form with the original expression ( x^2 - 6x + b ). By equating the coefficients of corresponding terms, I set up the following equations:1. The coefficient of ( x ) gives ( -2a = -6 ), which simplifies to ( a = 3 ).2. The constant term gives ( a^2 - 1 = b ). Substituting ( a = 3 ) into this equation, I find ( b = 9 - 1 = 8 ).Finally, to determine ( b - a ), I subtract ( a ) from ( b ): ( 8 - 3 = 5 ).

answer：First, I will list the given sequence: 0, 3, 8, 15, 24, 35,     , …Next, I will calculate the differences between consecutive terms to identify any patterns.The differences are:3 - 0 = 38 - 3 = 515 - 8 = 724 - 15 = 935 - 24 = 11I notice that the differences are increasing by 2 each time: 3, 5, 7, 9, 11. This suggests that the sequence is based on a quadratic relationship.To confirm this, I will express each term in terms of its position in the sequence. Let's denote the nth term as ( a_n ).Observing the pattern, it appears that:( a_n = n^2 - 1 )Let's verify this formula with the given terms:For ( n = 1 ): ( 1^2 - 1 = 0 )For ( n = 2 ): ( 2^2 - 1 = 3 )For ( n = 3 ): ( 3^2 - 1 = 8 )For ( n = 4 ): ( 4^2 - 1 = 15 )For ( n = 5 ): ( 5^2 - 1 = 24 )For ( n = 6 ): ( 6^2 - 1 = 35 )The formula holds true for all given terms. Therefore, to find the next term (( a_7 )):( a_7 = 7^2 - 1 = 49 - 1 = 48 )Thus, the number that should fill in the blank is 48.