💡First, I need to determine how many roses Bill buys and sells in each operation. He buys bouquets of 7 roses for 20 each and sells bouquets of 5 roses for 20 each. To make the process efficient, I'll consider buying 5 bouquets and selling them as 7 bouquets.Next, I'll calculate the total cost of buying 5 bouquets:[5 text{ bouquets} times 20 = 100]Then, I'll calculate the revenue from selling 7 bouquets:[7 text{ bouquets} times 20 = 140]The profit from one operation is the difference between the revenue and the cost:[140 - 100 = 40]To earn a total profit of 1000, I'll determine how many operations are needed:[frac{1000}{40} = 25 text{ operations}]Finally, since each operation involves buying 5 bouquets, the total number of bouquets Bill needs to buy is:[5 text{ bouquets/operation} times 25 text{ operations} = 125 text{ bouquets}]

💡Okay, so I need to find the area bounded by the curve ( y^2 = 9x ), the vertical lines ( x = 16 ) and ( x = 25 ), and the horizontal line ( y = 0 ). Hmm, let me think about how to approach this step by step.First, I remember that when dealing with areas bounded by curves and lines, integration is usually the way to go. Since the equation is given in terms of ( y^2 ), it might be easier to solve for ( y ) in terms of ( x ) so I can set up the integral properly.Starting with the equation ( y^2 = 9x ), if I take the square root of both sides, I get ( y = pm 3sqrt{x} ). But since we're dealing with an area bounded by ( y = 0 ), which is the x-axis, I think I only need to consider the upper half of the curve where ( y ) is positive. So, ( y = 3sqrt{x} ) is the function I'll work with.Now, I need to set up the integral to find the area between ( x = 16 ) and ( x = 25 ). The area under a curve from ( a ) to ( b ) is given by the integral of the function from ( a ) to ( b ). So, in this case, the area ( S ) should be:[S = int_{16}^{25} 3sqrt{x} , dx]Before I proceed, I should make sure I understand why I'm using this integral. The function ( y = 3sqrt{x} ) is the upper boundary, and ( y = 0 ) is the lower boundary. So, the vertical slices between ( x = 16 ) and ( x = 25 ) will give me the area when integrated.Next, I need to compute this integral. Integrating ( 3sqrt{x} ) with respect to ( x ). I recall that the integral of ( x^n ) is ( frac{x^{n+1}}{n+1} ), so applying that here:First, rewrite ( sqrt{x} ) as ( x^{1/2} ):[int 3x^{1/2} , dx]Now, integrate term by term:[3 cdot frac{x^{1/2 + 1}}{1/2 + 1} = 3 cdot frac{x^{3/2}}{3/2}]Simplify the expression:[3 div frac{3}{2} = 2]So, the integral simplifies to:[2x^{3/2}]Now, I need to evaluate this from ( x = 16 ) to ( x = 25 ):[S = left[ 2x^{3/2} right]_{16}^{25}]Calculating the upper limit first:[2 cdot 25^{3/2}]I know that ( 25^{1/2} = 5 ), so ( 25^{3/2} = 25^{1/2} cdot 25^{1} = 5 cdot 25 = 125 ). Therefore:[2 cdot 125 = 250]Now, calculating the lower limit:[2 cdot 16^{3/2}]Similarly, ( 16^{1/2} = 4 ), so ( 16^{3/2} = 4 cdot 16 = 64 ). Therefore:[2 cdot 64 = 128]Subtracting the lower limit from the upper limit gives:[250 - 128 = 122]So, the area bounded by the curve ( y^2 = 9x ), the lines ( x = 16 ), ( x = 25 ), and ( y = 0 ) is 122 square units.Wait, let me double-check my calculations to make sure I didn't make any mistakes. Starting from the integral:[int_{16}^{25} 3sqrt{x} , dx = left[ 2x^{3/2} right]_{16}^{25}]Calculating ( 25^{3/2} ):[25^{1/2} = 5, quad 5^3 = 125]So, ( 2 cdot 125 = 250 ).Calculating ( 16^{3/2} ):[16^{1/2} = 4, quad 4^3 = 64]So, ( 2 cdot 64 = 128 ).Subtracting:[250 - 128 = 122]Yes, that seems correct. I think I did everything right. The area is indeed 122 square units.

💡First, I need to determine Ram's initial deposit, which I'll denote as ( P ).At the end of the first year, the account accrued 100 in interest, bringing the total balance to 1,100. This means that:[P + 100 = 1100]Solving for ( P ):[P = 1100 - 100 = 1000]In the second year, the account balance increased by 20%. Therefore, the balance at the end of the second year is:[1100 times 1.20 = 1320]We are also told that at the end of the two years, the account balance increased by 32% from the initial deposit. This means:[P times 1.32 = 1320]Substituting ( P = 1000 ) into the equation:[1000 times 1.32 = 1320][1320 = 1320]This confirms that the initial deposit was indeed 1,000.

💡Let ( x ) be the total number of pages in the book.1. **First day**: Hui reads ( frac{1}{6}x + 10 ) pages. The remaining pages are: [ x - left(frac{1}{6}x + 10right) = frac{5}{6}x - 10 ]2. **Second day**: She reads ( frac{1}{5}left(frac{5}{6}x - 10right) + 20 ). The remaining pages after this day are: [ left(frac{5}{6}x - 10right) - left(frac{1}{5}left(frac{5}{6}x - 10right) + 20right) = frac{4}{5}left(frac{5}{6}x - 10right) - 20 = frac{2x}{3} - 32 ]3. **Third day**: She reads ( frac{1}{4}left(frac{2x}{3} - 32right) + 25 ). This leaves: [ left(frac{2x}{3} - 32right) - left(frac{1}{4}left(frac{2x}{3} - 32right) + 25right) = frac{3}{4}left(frac{2x}{3} - 32right) - 25 = frac{1}{2}x - 49 ]4. **Remaining pages**: Hui finds she has exactly ( 100 ) pages left: [ frac{1}{2}x - 49 = 100 implies frac{1}{2}x = 149 implies x = 298 ]Therefore, the total number of pages in the book is ( 298 ).

💡Okay, so I have this problem here: I need to find the value of ( k ) such that the polynomial ( 2x^3 - 8x^2 + kx - 10 ) is divisible by ( x - 2 ). The options are from A to E, with ( k ) being 12, 13, 14, 15, or 16. Hmm, let me think about how to approach this.First, I remember that if a polynomial is divisible by ( x - a ), then by the Remainder Theorem, substituting ( x = a ) into the polynomial should give zero. So in this case, since we're dealing with ( x - 2 ), I should substitute ( x = 2 ) into the polynomial and set it equal to zero. That should help me solve for ( k ).Alright, let me write that down. The polynomial is ( 2x^3 - 8x^2 + kx - 10 ). If I substitute ( x = 2 ), it should equal zero. So plugging in 2 for x:( 2(2)^3 - 8(2)^2 + k(2) - 10 = 0 )Let me compute each term step by step. First, ( 2(2)^3 ). ( 2^3 ) is 8, so 2 times 8 is 16.Next term: ( -8(2)^2 ). ( 2^2 ) is 4, so -8 times 4 is -32.Third term: ( k(2) ) is just 2k.Last term: -10.Putting it all together: 16 - 32 + 2k - 10 = 0.Now, let's simplify this equation. 16 - 32 is -16, and then -16 - 10 is -26. So the equation becomes:-26 + 2k = 0To solve for ( k ), I can add 26 to both sides:2k = 26Then, divide both sides by 2:k = 13So, ( k ) should be 13. Let me check the options: option B is 13, so that should be the correct answer.Wait, just to make sure I didn't make any calculation errors, let me go through the steps again.Substituting ( x = 2 ):( 2(2)^3 = 2*8 = 16 )( -8(2)^2 = -8*4 = -32 )( k(2) = 2k )( -10 )Adding them up: 16 - 32 = -16; -16 + 2k -10 = -26 + 2k.Set equal to zero: -26 + 2k = 0 => 2k = 26 => k = 13.Yep, that seems consistent. Maybe I can also try polynomial division to confirm.If I perform polynomial division of ( 2x^3 - 8x^2 + 13x - 10 ) by ( x - 2 ), I should get zero remainder.Let me set it up:Divide ( 2x^3 - 8x^2 + 13x - 10 ) by ( x - 2 ).First term: ( 2x^3 ) divided by ( x ) is ( 2x^2 ). Multiply ( 2x^2 ) by ( x - 2 ) gives ( 2x^3 - 4x^2 ).Subtract that from the original polynomial:( (2x^3 - 8x^2 + 13x - 10) - (2x^3 - 4x^2) = (-8x^2 + 4x^2) + 13x -10 = -4x^2 + 13x -10 ).Next term: ( -4x^2 ) divided by ( x ) is ( -4x ). Multiply ( -4x ) by ( x - 2 ) gives ( -4x^2 + 8x ).Subtract that from the current polynomial:( (-4x^2 + 13x -10) - (-4x^2 + 8x) = (13x - 8x) -10 = 5x -10 ).Next term: ( 5x ) divided by ( x ) is 5. Multiply 5 by ( x - 2 ) gives ( 5x -10 ).Subtract that:( (5x -10) - (5x -10) = 0 ).So the remainder is zero, which confirms that when ( k = 13 ), the polynomial is indeed divisible by ( x - 2 ). That solidifies that the correct answer is 13, which is option B.I think that's thorough enough. I used the Remainder Theorem to find ( k ), then double-checked by performing polynomial division. Both methods led me to the same conclusion, so I'm confident that ( k = 13 ) is correct.**Final Answer**boxed{B}

💡Okay, so I need to find the expected value of rolling a twelve-sided die that's numbered from 1 to 12. Hmm, I remember that expected value is like the average outcome you'd expect if you rolled the die many times. But I'm not entirely sure how to calculate it step by step. Let me think this through.First, I know that a twelve-sided die has numbers from 1 to 12. So, there are 12 possible outcomes when you roll it. Each outcome is equally likely, right? That means each number has the same chance of coming up, which is 1 out of 12. So, the probability for each number is 1/12.Now, to find the expected value, I think I need to multiply each possible outcome by its probability and then add all those products together. That sounds like the formula for expected value: E = Σ (x_i * P(x_i)), where x_i are the outcomes and P(x_i) is the probability of each outcome.So, applying that here, I need to calculate the sum of all numbers from 1 to 12, each multiplied by 1/12. That seems straightforward, but adding all those numbers individually would take time. Maybe there's a quicker way to find the sum of numbers from 1 to 12.Wait, I recall there's a formula for the sum of the first n natural numbers. It's n(n + 1)/2. Let me check that. If n is 12, then the sum should be 12*13/2, which is 78. Yeah, that makes sense because 12 times 13 is 156, and divided by 2 is 78.So, the total sum of all outcomes is 78. Now, since there are 12 outcomes, the average or expected value would be the total sum divided by the number of outcomes. That would be 78 divided by 12. Let me do that division: 78 divided by 12 is 6.5.Wait, 6.5 is a decimal, but expected value can be a decimal, right? It doesn't have to be a whole number because it's an average. So, 6.5 seems reasonable.Let me double-check my steps to make sure I didn't make a mistake. I identified the number of outcomes correctly as 12, each with equal probability of 1/12. Then, I used the formula for the sum of the first n natural numbers to find the total sum, which is 78. Dividing that by 12 gives me 6.5. Yeah, that all adds up.I guess another way to think about it is that the numbers are evenly distributed from 1 to 12, so the average should be right in the middle. The middle of 1 and 12 is (1 + 12)/2, which is also 6.5. That confirms my earlier calculation.So, I'm pretty confident that the expected value of rolling a twelve-sided die numbered from 1 to 12 is 6.5.