answer：First, I need to determine the possible configurations of the isosceles triangle with side lengths 5 and 8.In an isosceles triangle, two sides are equal. There are two possible cases:1. **Case 1:** The two equal sides are both 5 units long. - The third side is 8 units. - Perimeter = 5 + 5 + 8 = 18 units.2. **Case 2:** The two equal sides are both 8 units long. - The third side is 5 units. - Perimeter = 8 + 8 + 5 = 21 units.Therefore, the possible perimeters of the isosceles triangle are 18 units or 21 units.

answer：Okay, so I have this problem about a store selling a disinfectant product. They bought it for 8 yuan each, and they noticed that the daily sales quantity, which is y, has a linear relationship with the selling price per item, which is x. The price x is between 8 and 15 yuan, and it's an integer. They gave me two specific points: when the price is 9 yuan, they sold 105 items, and when the price is 11 yuan, they sold 95 items. I need to find the functional relationship between y and x first. Then, I have to figure out what price they should set to maximize their daily profit, and what that maximum profit would be.Alright, starting with part (1). Since it's a linear relationship, I can model it with a straight line equation, which is usually y = mx + b, where m is the slope and b is the y-intercept. But here, y is the quantity sold, and x is the price. So, it's more like y = kx + c, where k is the slope and c is the intercept.They gave me two points: (9, 105) and (11, 95). So, I can use these two points to find the slope k. The formula for the slope between two points (x1, y1) and (x2, y2) is (y2 - y1)/(x2 - x1). Let me calculate that.So, y2 is 95, y1 is 105, x2 is 11, and x1 is 9. Plugging in the numbers: (95 - 105)/(11 - 9) = (-10)/2 = -5. So, the slope k is -5. That means for every 1 yuan increase in price, the quantity sold decreases by 5 items.Now that I have the slope, I can find the intercept c. I'll use one of the given points to plug into the equation y = kx + c. Let's use (9, 105). So, 105 = -5*9 + c. Calculating that: -5*9 is -45, so 105 = -45 + c. Adding 45 to both sides gives c = 150.So, the equation is y = -5x + 150. Let me check if this works with the other point. If x is 11, then y should be -5*11 + 150 = -55 + 150 = 95. Yep, that matches. So, part (1) is done.Moving on to part (2). They want to maximize the daily profit. Profit is usually calculated as (selling price - cost price) * number of items sold. The cost price is 8 yuan per item, and the selling price is x yuan. The number of items sold is y, which we have as a function of x: y = -5x + 150.So, profit w can be written as w = (x - 8) * y. Substituting y, we get w = (x - 8)*(-5x + 150). Let me expand this equation.First, multiply (x - 8) with (-5x + 150). Using the distributive property:w = x*(-5x + 150) - 8*(-5x + 150)w = -5x² + 150x + 40x - 1200w = -5x² + 190x - 1200So, the profit function is a quadratic equation: w = -5x² + 190x - 1200. Since the coefficient of x² is negative (-5), the parabola opens downward, meaning the vertex is the maximum point. The vertex of a parabola given by ax² + bx + c is at x = -b/(2a).Let me compute that. Here, a = -5 and b = 190. So, x = -190/(2*(-5)) = -190/(-10) = 19. So, the maximum profit occurs at x = 19 yuan.But wait, the problem states that x is between 8 and 15, and x is an integer. 19 is outside the given range. That means the maximum profit within the allowed range isn't at the vertex but at the highest possible x within the range. So, since x can go up to 15, I should check the profit at x = 15.Let me calculate the profit at x = 15. Using the profit function w = -5x² + 190x - 1200:w = -5*(15)² + 190*15 - 1200w = -5*225 + 2850 - 1200w = -1125 + 2850 - 1200w = (2850 - 1125) - 1200w = 1725 - 1200w = 525So, at x = 15, the profit is 525 yuan. Let me check the profit at x = 14 to make sure it's indeed lower. w = -5*(14)² + 190*14 - 1200w = -5*196 + 2660 - 1200w = -980 + 2660 - 1200w = (2660 - 980) - 1200w = 1680 - 1200w = 480Yes, 480 is less than 525. Similarly, at x = 16, which is beyond the allowed range, but just to check:w = -5*(16)² + 190*16 - 1200w = -5*256 + 3040 - 1200w = -1280 + 3040 - 1200w = (3040 - 1280) - 1200w = 1760 - 1200w = 560But 16 is outside the allowed range, so we can't consider it. So, within 8 to 15, the maximum profit is at x = 15, which is 525 yuan.Wait, just to make sure, let me check x = 15 and x = 14 again. At x = 15, y = -5*15 + 150 = -75 + 150 = 75 items sold. So, profit is (15 - 8)*75 = 7*75 = 525. Yep, that's correct.At x = 14, y = -5*14 + 150 = -70 + 150 = 80 items sold. Profit is (14 - 8)*80 = 6*80 = 480. Correct.So, it seems that as the price increases, the profit increases up to a point, but since the maximum allowed price is 15, that's where the profit peaks within the given constraints.Therefore, the store should set the price at 15 yuan per item to maximize the daily profit, which would be 525 yuan.**Final Answer**(1) The functional relationship is boxed{y = -5x + 150}.(2) The disinfectant product should be sold at boxed{15} yuan per item to maximize the daily profit, and the maximum profit is boxed{525} yuan.

answer：First, identify the standard distance for the long jump competition, which is (4.00) meters.Next, calculate the difference between Xiao Dong's jump distance and the standard distance for each jump.For the first jump:- Xiao Dong's jump distance = (4.22) meters- Standard distance = (4.00) meters- Difference = (4.22 - 4.00 = +0.22) metersFor the second jump:- Xiao Dong's jump distance = (3.85) meters- Standard distance = (4.00) meters- Difference = (3.85 - 4.00 = -0.15) metersThe negative sign indicates that the jump is shorter than the standard distance.

answer：Okay, so I have this problem where I need to find a 2x2 matrix N such that when I multiply it by the vector [2, 1], I get [5, 4], and when I multiply it by [1, -4], I get [0, -9]. Hmm, that sounds like a system of equations problem. Let me think about how to approach this.First, I remember that a matrix multiplied by a vector is just a linear combination of the columns of the matrix. So, if N is a 2x2 matrix, let's denote it as:N = [ [a, b], [c, d] ]Then, when I multiply N by [2, 1], it should be equal to [5, 4]. So, that gives me:2a + 1b = 52c + 1d = 4Similarly, when I multiply N by [1, -4], it should be equal to [0, -9]. So, that gives me:1a + (-4)b = 01c + (-4)d = -9So now I have four equations:1. 2a + b = 52. 2c + d = 43. a - 4b = 04. c - 4d = -9Alright, so I have four equations with four variables: a, b, c, d. I need to solve this system.Let me start with the first pair of equations:From equation 3: a = 4bSo, I can substitute a in equation 1:2*(4b) + b = 58b + b = 59b = 5b = 5/9Then, a = 4b = 4*(5/9) = 20/9Okay, so a = 20/9 and b = 5/9.Now, moving on to the second pair of equations:From equation 4: c = 4d - 9Substitute c into equation 2:2*(4d - 9) + d = 48d - 18 + d = 49d - 18 = 49d = 22d = 22/9Then, c = 4d - 9 = 4*(22/9) - 9 = 88/9 - 81/9 = 7/9So, c = 7/9 and d = 22/9.Putting it all together, the matrix N is:[ [20/9, 5/9], [7/9, 22/9] ]Let me double-check to make sure this works.First, multiply N by [2, 1]:First component: 2*(20/9) + 1*(5/9) = 40/9 + 5/9 = 45/9 = 5Second component: 2*(7/9) + 1*(22/9) = 14/9 + 22/9 = 36/9 = 4Good, that gives [5, 4].Now, multiply N by [1, -4]:First component: 1*(20/9) + (-4)*(5/9) = 20/9 - 20/9 = 0Second component: 1*(7/9) + (-4)*(22/9) = 7/9 - 88/9 = (-81)/9 = -9Perfect, that gives [0, -9].So, the matrix N is correct.**Final Answer**The matrix (mathbf{N}) is boxed{begin{pmatrix} dfrac{20}{9} & dfrac{5}{9} dfrac{7}{9} & dfrac{22}{9} end{pmatrix}}.

answer：Okay, so I have this problem: In triangle ABC, if the sides a, b, c form a geometric sequence, then I need to find the value of cos 2B + cos B + cos(A - C). Hmm, interesting. Let me try to figure this out step by step.First, let me recall what a geometric sequence is. In a geometric sequence, each term after the first is found by multiplying the previous term by a constant called the common ratio. So, if a, b, c are in a geometric sequence, then b/a = c/b, which implies that b² = a*c. That's a key relationship I can use here.Now, in triangle ABC, the sides are related to the angles opposite them through the Law of Sines. The Law of Sines states that a/sin A = b/sin B = c/sin C = 2R, where R is the circumradius of the triangle. So, I can express each side in terms of the sine of the opposite angle and the circumradius.Let me write that down:a = 2R sin Ab = 2R sin Bc = 2R sin CSince a, b, c form a geometric sequence, we have b² = a*c. Substituting the expressions from the Law of Sines into this equation, we get:(2R sin B)² = (2R sin A)(2R sin C)Simplifying this, we have:4R² sin² B = 4R² sin A sin CDividing both sides by 4R², we get:sin² B = sin A sin COkay, so that's another important relationship: sin² B = sin A sin C.Now, the problem asks for cos 2B + cos B + cos(A - C). Let me break this down into parts.First, let's recall the double-angle identity for cosine: cos 2B = 2 cos² B - 1. So, I can rewrite cos 2B as 2 cos² B - 1.So, the expression becomes:2 cos² B - 1 + cos B + cos(A - C)Now, let's see if I can find expressions for cos(A - C) and relate it to the other terms.I know that in a triangle, the sum of the angles is π radians (or 180 degrees). So, A + B + C = π. Therefore, A + C = π - B.Hmm, that might be useful. Let me think about how to express cos(A - C). Maybe using the sum and difference formulas?Yes, the cosine of a difference can be expressed as:cos(A - C) = cos A cos C + sin A sin CSo, substituting that into our expression, we have:2 cos² B - 1 + cos B + cos A cos C + sin A sin CNow, from earlier, we have sin² B = sin A sin C. So, sin A sin C = sin² B.Therefore, substituting that in, we get:2 cos² B - 1 + cos B + cos A cos C + sin² BNow, let's recall that sin² B + cos² B = 1. So, sin² B = 1 - cos² B.Substituting that into the expression, we have:2 cos² B - 1 + cos B + cos A cos C + (1 - cos² B)Simplifying this, the -1 and +1 cancel out:2 cos² B + cos B + cos A cos C - cos² BWhich simplifies to:cos² B + cos B + cos A cos CHmm, so now I have:cos² B + cos B + cos A cos CI need to find a way to relate cos A cos C to something else. Maybe using the fact that A + C = π - B?Yes, let's consider that. Since A + C = π - B, we can use the cosine of sum formula:cos(A + C) = cos(π - B) = -cos BBut cos(A + C) can also be expanded as:cos A cos C - sin A sin C = -cos BFrom earlier, we have sin A sin C = sin² B, so substituting that in:cos A cos C - sin² B = -cos BTherefore, cos A cos C = sin² B - cos BBut sin² B = 1 - cos² B, so substituting that in:cos A cos C = (1 - cos² B) - cos B = 1 - cos² B - cos BSo, cos A cos C = 1 - cos² B - cos BNow, going back to our expression:cos² B + cos B + cos A cos CSubstituting cos A cos C:cos² B + cos B + (1 - cos² B - cos B)Simplifying this:cos² B + cos B + 1 - cos² B - cos BThe cos² B and -cos² B cancel out, and the cos B and -cos B cancel out, leaving us with:1So, the entire expression simplifies to 1.Wait, let me double-check that to make sure I didn't make a mistake.Starting from:cos 2B + cos B + cos(A - C) = 2 cos² B - 1 + cos B + cos(A - C)Then, cos(A - C) = cos A cos C + sin A sin CWe know sin A sin C = sin² B, so:= 2 cos² B - 1 + cos B + cos A cos C + sin² BThen, sin² B = 1 - cos² B, so:= 2 cos² B - 1 + cos B + cos A cos C + 1 - cos² BSimplify:2 cos² B - 1 + 1 - cos² B + cos B + cos A cos CWhich is:cos² B + cos B + cos A cos CThen, from earlier, cos A cos C = 1 - cos² B - cos BSo, substituting:cos² B + cos B + 1 - cos² B - cos B = 1Yes, that seems correct. So, the entire expression simplifies to 1.Therefore, the value of cos 2B + cos B + cos(A - C) is 1.**Final Answer**boxed{1}

answer：First, I need to understand the given ratio of boys to girls, which is 8:5. This means that for every 8 boys, there are 5 girls.Next, I know there are 210 girls in the college. Using the ratio, I can determine how many "parts" each part of the ratio represents. Since 5 parts correspond to 210 girls, I can calculate the number of students per part by dividing 210 by 5, which gives me 42 students per part.Now, to find the number of boys, I multiply the number of parts for boys (8) by the number of students per part (42). This gives me 336 boys.Finally, to find the total number of students, I add the number of boys and girls together: 336 boys + 210 girls = 546 students.