💡To determine if it's possible to end up with 2007 small pieces of paper after repeatedly tearing a large piece of paper into four pieces each time, we can analyze the process step by step.1. **Initial Setup:** - We start with 1 large piece of paper.2. **First Tear:** - After tearing the paper into four pieces, we have 4 pieces.3. **Subsequent Tears:** - Each time we tear one piece into four, we increase the total number of pieces by 3 (since one piece is replaced by four, resulting in a net gain of 3 pieces).4. **General Formula:** - Let ( n ) be the number of tears. - The total number of pieces after ( n ) tears can be expressed as: [ 1 + 3n ]5. **Checking for 2007 Pieces:** - We need to solve the equation: [ 1 + 3n = 2007 ] - Simplifying: [ 3n = 2006 ] [ n = frac{2006}{3} ] - Calculating: [ frac{2006}{3} = 668.666ldots ] - Since ( n ) must be an integer, and 668.666... is not an integer, it's impossible to achieve exactly 2007 pieces.Thus, it is impossible to obtain exactly 2007 pieces of paper through this tearing process.

💡First, I need to determine the interval in which the zero of the function ( y = ln x - 6 + 2x ) lies. To do this, I'll evaluate the function at the endpoints of each given interval.Starting with interval A: (1, 2)- At ( x = 1 ): [ y = ln 1 - 6 + 2(1) = 0 - 6 + 2 = -4 ] The function value is negative.- At ( x = 2 ): [ y = ln 2 - 6 + 2(2) = ln 2 - 6 + 4 = ln 2 - 2 approx 0.6931 - 2 = -1.3069 ] The function value is still negative. Since the function does not change sign in this interval, the zero is not in (1, 2).Next, interval B: (2, 3)- At ( x = 2 ): [ y = ln 2 - 6 + 4 = ln 2 - 2 approx -1.3069 ] The function value is negative.- At ( x = 3 ): [ y = ln 3 - 6 + 6 = ln 3 approx 1.0986 ] The function value is positive. Since the function changes sign from negative to positive between ( x = 2 ) and ( x = 3 ), the zero must lie within this interval.For completeness, I'll check the remaining intervals to ensure there are no other zeros.- Interval C: (3, 4) - At ( x = 3 ): [ y = ln 3 approx 1.0986 ] Positive. - At ( x = 4 ): [ y = ln 4 - 6 + 8 = ln 4 + 2 approx 1.3863 + 2 = 3.3863 ] Positive. No sign change. - Interval D: (5, 6) - At ( x = 5 ): [ y = ln 5 - 6 + 10 = ln 5 + 4 approx 1.6094 + 4 = 5.6094 ] Positive. - At ( x = 6 ): [ y = ln 6 - 6 + 12 = ln 6 + 6 approx 1.7918 + 6 = 7.7918 ] Positive. No sign change.Since the function changes sign only in interval B: (2, 3), the zero ( x_0 ) must lie within this interval.

💡To determine how many whole numbers from 1 through 60 are divisible by 4, 5, or both, we can use the principle of inclusion and exclusion.1. **Count numbers divisible by 4**: [ leftlfloor frac{60}{4} rightrfloor = 15 ] There are 15 numbers divisible by 4.2. **Count numbers divisible by 5**: [ leftlfloor frac{60}{5} rightrfloor = 12 ] There are 12 numbers divisible by 5.3. **Initial sum of numbers divisible by 4 or 5**: [ 15 + 12 = 27 ] However, this count includes numbers divisible by both 4 and 5 twice.4. **Count numbers divisible by both 4 and 5 (i.e., divisible by 20)**: The least common multiple (LCM) of 4 and 5 is 20. [ leftlfloor frac{60}{20} rightrfloor = 3 ] There are 3 numbers divisible by both 4 and 5.5. **Adjust for overcount**: [ 27 - 3 = 24 ] After adjusting for numbers accounted twice, the correct count of numbers divisible by 4 or 5 or both is 24.6. **Conclusion**: [ 24 ]The final answer is (boxed{text{B}} 24).

💡First, I need to understand the problem. Alice has the number 30, and Bob's number must include all the prime factors of 30.I'll start by finding the prime factors of 30. Breaking it down:[30 = 2 times 3 times 5]So, the prime factors are 2, 3, and 5.Bob's number must include each of these prime factors. To find the smallest possible number, I'll multiply these prime factors together:[2 times 3 times 5 = 30]Therefore, the smallest number Bob could have is 30.

💡Okay, so I have this function g(x) which is defined as 2x squared minus 3x plus 1. And I need to find the value of g(-2). Hmm, let me think about how to approach this.First, I remember that when you're given a function like this, to find the value at a specific point, you just substitute that value into the function wherever you see x. So in this case, I need to plug in -2 for x in the equation.Alright, let's write that out. So, g(-2) would be equal to 2 times (-2) squared minus 3 times (-2) plus 1. Let me make sure I got that right. Yeah, that seems correct.Now, I need to calculate each part step by step. Let's start with the first term: 2 times (-2) squared. I know that when you square a negative number, the result is positive because a negative times a negative is a positive. So, (-2) squared is 4. Then, multiplying that by 2 gives me 8. Okay, so the first term is 8.Next, the second term is minus 3 times (-2). Hmm, multiplying a negative by a negative should give a positive result. So, -3 times -2 is 6. That makes sense. So, the second term is 6.Now, the last term is just plus 1. That's straightforward. So, putting it all together, I have 8 from the first term, plus 6 from the second term, and then plus 1 from the last term.Adding those up: 8 plus 6 is 14, and then 14 plus 1 is 15. So, putting it all together, g(-2) equals 15.Wait, let me double-check my calculations to make sure I didn't make any mistakes. So, (-2) squared is 4, multiplied by 2 is 8. Then, -3 times -2 is 6. Adding those together with the 1 gives me 8 plus 6 is 14, plus 1 is 15. Yeah, that seems right.I don't think I made any errors here. Each step seems to follow logically, and the arithmetic checks out. So, I'm confident that the value of g(-2) is indeed 15.

💡Okay, so I need to figure out how many touchdowns the Seattle Seahawks scored in their game against the Denver Broncos. The final score was 37-23, and the Seahawks won. I know that a touchdown is worth 7 points and a field goal is worth 3 points. The problem also tells me that the Seahawks scored 3 field goals during the game.First, I should calculate how many points the Seahawks got from the field goals. Since each field goal is worth 3 points and they scored 3 of them, I can multiply 3 by 3 to find the total points from field goals.So, 3 field goals × 3 points per field goal = 9 points from field goals.Now, I know the total points the Seahawks scored was 37. If 9 of those points came from field goals, then the remaining points must have come from touchdowns. To find out how many points came from touchdowns, I'll subtract the points from field goals from the total points.37 total points - 9 points from field goals = 28 points from touchdowns.Since each touchdown is worth 7 points, I can find out how many touchdowns there were by dividing the points from touchdowns by the points per touchdown.28 points from touchdowns ÷ 7 points per touchdown = 4 touchdowns.So, the Seattle Seahawks scored 4 touchdowns in the game.