answer：First, I need to determine the cost of one bookmark. Both groups of students paid a total amount for their bookmarks, and the cost per bookmark must be a whole number of cents. I'll start by converting the total payments from dollars to cents to make the calculations easier:- The fifth graders paid 2.25, which is 225 cents.- The fourth graders paid 2.60, which is 260 cents.Next, I need to find a common factor of 225 and 260 that represents the cost per bookmark. I'll factorize both amounts:- 225 can be factored into (3^2 times 5^2).- 260 can be factored into (2^2 times 5 times 13).The common factor between 225 and 260 is 5. Therefore, the cost of one bookmark is 5 cents.Now, I'll calculate how many bookmarks each group bought:- The fifth graders bought (frac{225}{5} = 45) bookmarks.- The fourth graders bought (frac{260}{5} = 52) bookmarks.Finally, I'll find the difference in the number of bookmarks bought by the two groups:- (52 - 45 = 7).So, 7 more fourth graders than fifth graders bought bookmarks.

answer：Alright, so I have this problem here with two functions, f(x) and g(x), and I need to find the range of the real number a such that for any x1 and x2 in real numbers, f(x1) is less than or equal to g(x2). The options are given as A: [0,2], B: ℝ, C: [-2,0], D: (-∞, -2] ∪ [0, +∞). Hmm, okay, let me try to break this down step by step.First, let me understand the functions involved. The function f(x) is piecewise defined. For x less than or equal to 1, it's a quadratic function: -x² + 2x - 2. For x greater than 1, it's an exponential function: 2^{-|1 - x|} - 2. Okay, so f(x) has two different expressions depending on whether x is less than or equal to 1 or greater than 1.Then, there's g(x) = |a - 1| cos x. That's a cosine function scaled by the absolute value of (a - 1). So, the amplitude of the cosine function is |a - 1|.The condition given is that for any x1 and x2 in ℝ, f(x1) ≤ g(x2). So, no matter what x1 and x2 I pick, f evaluated at x1 should be less than or equal to g evaluated at x2. That means the maximum value of f(x) should be less than or equal to the minimum value of g(x). Because if the maximum of f is less than or equal to the minimum of g, then certainly for any x1 and x2, f(x1) will be less than or equal to g(x2).So, my first task is to find the maximum value of f(x). Let's tackle that.Starting with the first piece of f(x): for x ≤ 1, f(x) = -x² + 2x - 2. This is a quadratic function, and since the coefficient of x² is negative (-1), it opens downward, meaning it has a maximum at its vertex.The vertex of a quadratic function ax² + bx + c is at x = -b/(2a). So, here, a = -1, b = 2. So, x = -2/(2*(-1)) = -2/(-2) = 1. So, the vertex is at x = 1. Therefore, the maximum value of this quadratic occurs at x = 1.Let me compute f(1): f(1) = -(1)^2 + 2*(1) - 2 = -1 + 2 - 2 = -1. So, the maximum value of the first piece is -1.Now, let's look at the second piece of f(x): for x > 1, f(x) = 2^{-|1 - x|} - 2. Let me simplify this expression. The exponent is -|1 - x|. Since x > 1, |1 - x| = x - 1. So, 2^{-|1 - x|} = 2^{-(x - 1)} = 2^{1 - x}.Therefore, for x > 1, f(x) = 2^{1 - x} - 2. Now, let's analyze this function. 2^{1 - x} is an exponential decay function because the exponent is negative. As x increases, 2^{1 - x} approaches 0. So, as x approaches infinity, f(x) approaches -2.At x = 1, let's compute the value: f(1) = 2^{1 - 1} - 2 = 2^0 - 2 = 1 - 2 = -1. So, at x = 1, both pieces of f(x) meet at the point (1, -1). That's good; the function is continuous at x = 1.Now, for x > 1, f(x) starts at -1 when x = 1 and decreases towards -2 as x increases. So, the maximum value of the second piece is also -1 at x = 1.Therefore, the maximum value of the entire function f(x) is -1.Okay, so f_max = -1.Now, moving on to g(x) = |a - 1| cos x. The cosine function oscillates between -1 and 1. So, when we scale it by |a - 1|, the range of g(x) becomes from -|a - 1| to |a - 1|.But in our condition, we need f(x1) ≤ g(x2) for any x1, x2. Since f_max is -1, we need -1 ≤ g(x2) for all x2. But wait, g(x2) can be as low as -|a - 1| and as high as |a - 1|.But to satisfy f(x1) ≤ g(x2) for all x1, x2, we need the maximum of f(x1) to be less than or equal to the minimum of g(x2). Because if the maximum of f is less than or equal to the minimum of g, then certainly f(x1) will always be less than or equal to g(x2).So, the minimum value of g(x) is -|a - 1|. Therefore, we need:f_max ≤ g_minWhich translates to:-1 ≤ -|a - 1|Let me solve this inequality.-1 ≤ -|a - 1|Multiply both sides by -1. Remember that multiplying both sides of an inequality by a negative number reverses the inequality sign.So:1 ≥ |a - 1|Which is the same as:|a - 1| ≤ 1This means that the distance between a and 1 on the real number line is at most 1. So, a must lie within the interval [1 - 1, 1 + 1], which is [0, 2].Therefore, the range of a is [0, 2], which corresponds to option A.Wait, let me double-check my reasoning. I found that f_max is -1, and g_min is -|a - 1|. So, to have f_max ≤ g_min, we have -1 ≤ -|a - 1|, which simplifies to |a - 1| ≤ 1, leading to a ∈ [0, 2]. That seems correct.Alternatively, if I think about it differently, since g(x) = |a - 1| cos x, the minimum value of g(x) is -|a - 1|. So, for f(x1) ≤ g(x2) for all x1, x2, the maximum of f(x1) must be ≤ the minimum of g(x2). So, yes, -1 ≤ -|a - 1|, which again gives |a - 1| ≤ 1, so a ∈ [0, 2].I think that's solid. So, the answer should be A: [0,2].**Final Answer**boxed{A}

answer：First, I recognize that Ahmed is currently 11 years old and Fouad is 26 years old. The goal is to determine in how many years Fouad's age will be a multiple of Ahmed's current age.Let’s denote the number of years into the future when this condition is met as ( x ).In ( x ) years, Fouad's age will be ( 26 + x ).We need ( 26 + x ) to be a multiple of 11. This means that ( 26 + x ) should be equal to ( 11k ), where ( k ) is an integer.To find the smallest such ( x ), I start by dividing 26 by 11 to find the next whole number multiple:[26 div 11 = 2 text{ with a remainder of } 4]The next multiple of 11 after 26 is ( 3 times 11 = 33 ).Setting up the equation:[26 + x = 33]Solving for ( x ):[x = 33 - 26 = 7]In 7 years, Fouad will be 33 years old, which is a multiple of Ahmed's current age (11). The multiple is:[33 div 11 = 3]Therefore, the multiple of Ahmed's age that Fouad's age will be in that number of years is 3.

answer：First, I need to calculate the product of 1.6 and 0.5. Multiplying these two numbers gives me 0.8.Next, I have to divide this product by 1. Dividing any number by 1 leaves the number unchanged, so 0.8 divided by 1 is still 0.8.Therefore, the final result is 0.8.

answer：First, I need to determine the distance driven on local roads, which I'll denote as ( x ) miles.The time taken to drive on local roads at 30 mph is ( frac{x}{30} ) hours.The distance driven on the highway is 65 miles, and the speed on the highway is 65 mph. Therefore, the time taken to drive on the highway is ( frac{65}{65} = 1 ) hour.The total distance driven is ( x + 65 ) miles, and the total time taken is ( frac{x}{30} + 1 ) hours.The average speed for the entire trip is given as 41.67 mph. The average speed is calculated by dividing the total distance by the total time:[text{Average speed} = frac{text{Total distance}}{text{Total time}}]Substituting the given average speed and the expressions for total distance and total time, I get:[41.67 = frac{x + 65}{frac{x}{30} + 1}]Next, I solve for ( x ):[41.67 left( frac{x}{30} + 1 right) = x + 65][41.67 cdot frac{x}{30} + 41.67 = x + 65][frac{41.67x}{30} + 41.67 = x + 65]To clear the fraction, I multiply every term by 30:[41.67x + 1250.1 = 30x + 1950]Subtracting ( 30x ) from both sides:[11.67x + 1250.1 = 1950]Subtracting ( 1250.1 ) from both sides:[11.67x = 699.9]Finally, dividing by ( 11.67 ) to solve for ( x ):[x = frac{699.9}{11.67}][x approx 59.97]So, the car drove approximately 60 miles on local roads.

answer：Okay, so I need to find the length of this parametric curve given by (x, y) = (3 sin t, 3 cos t) from t = 0 to t = 2π. Hmm, parametric curves... I remember they're defined by equations where both x and y are expressed in terms of a third variable, usually t in this case. First, I think I should recall the formula for the length of a parametric curve. I believe it involves an integral that sums up the infinitesimal lengths along the curve. Let me try to remember. I think it's something like the integral from t = a to t = b of the square root of (dx/dt)^2 + (dy/dt)^2 dt. Yeah, that sounds right. So, the formula is:L = ∫₀²π √[(dx/dt)² + (dy/dt)²] dtAlright, so I need to find dx/dt and dy/dt. Let's compute those derivatives.Given x = 3 sin t, so dx/dt would be the derivative of 3 sin t with respect to t. The derivative of sin t is cos t, so dx/dt = 3 cos t.Similarly, y = 3 cos t, so dy/dt is the derivative of 3 cos t with respect to t. The derivative of cos t is -sin t, so dy/dt = -3 sin t.Now, I need to square both dx/dt and dy/dt and add them together. Let's compute that:(dx/dt)² = (3 cos t)² = 9 cos² t(dy/dt)² = (-3 sin t)² = 9 sin² tAdding them together: 9 cos² t + 9 sin² t = 9 (cos² t + sin² t)Wait, I remember that cos² t + sin² t = 1, that's one of the Pythagorean identities. So, this simplifies to 9 * 1 = 9.So, the integrand becomes √9 = 3.Therefore, the integral for the length simplifies to:L = ∫₀²π 3 dtThat's much easier to integrate. The integral of 3 with respect to t is just 3t. So, evaluating from 0 to 2π:L = [3t]₀²π = 3*(2π) - 3*(0) = 6π - 0 = 6πSo, the length of the parametric curve is 6π.Wait a second, let me think if there's another way to see this. The parametric equations given are x = 3 sin t and y = 3 cos t. If I square both x and y and add them, I get:x² + y² = (3 sin t)² + (3 cos t)² = 9 sin² t + 9 cos² t = 9 (sin² t + cos² t) = 9*1 = 9So, x² + y² = 9, which is the equation of a circle with radius 3 centered at the origin. Therefore, the curve is a circle of radius 3, and the length of the curve from t = 0 to t = 2π is just the circumference of the circle.The circumference of a circle is 2πr, so in this case, it's 2π*3 = 6π. That matches what I got earlier using the parametric arc length formula. So, that's a good consistency check.I think I'm confident now that the length of the curve is 6π.