💡To find the length of the train, I need to determine the distance it travels while crossing the pole. The train's speed is given in kilometers per hour, but the time is in seconds. To ensure consistency in units, I'll convert the speed from km/hr to meters per second (m/s).First, I'll convert 180 km/hr to m/s:180 km/hr × (1000 meters / 3600 seconds) = 50 m/sNow that the speed is in meters per second, I can calculate the distance traveled in 18 seconds:Distance = Speed × Time = 50 m/s × 18 s = 900 metersTherefore, the length of the train is 900 meters.

💡First, I need to find the value of ( m ) by substituting the pair ((-2, 3)) into the formula ( a^2 + b + 1 ).Calculating ( m ):[m = (-2)^2 + 3 + 1 = 4 + 3 + 1 = 8]Next, I use the value of ( m ) which is 8 and substitute it along with 1 into the same formula to find the final result.Calculating the final result:[8^2 + 1 + 1 = 64 + 1 + 1 = 66]Therefore, the resulting rational number is 66.

💡Alright, so I have this equation: √x + √y = √1960. I need to find how many integer solutions there are for x and y. Hmm, okay, let's break this down step by step.First, I know that √1960 is a bit of a big number. Maybe I should simplify that to make things easier. Let me see, 1960. Hmm, 1960 divided by 16 is 122.5, which isn't a perfect square. Wait, maybe I can factor 1960 into something that has a perfect square. Let's try that.1960. I know that 196 is 14 squared, right? So 1960 is 196 times 10. So, √1960 is √(196*10), which is √196 * √10, which is 14√10. Okay, so √1960 simplifies to 14√10. So now, the equation becomes √x + √y = 14√10.Hmm, so I have √x + √y = 14√10. I need to find integer solutions for x and y. That means both x and y must be perfect squares because their square roots are being added together. So, let me think about how to express 14√10 as a sum of two square roots.Let me denote √x as a and √y as b. So, the equation becomes a + b = 14√10. Since a and b are square roots of integers, they must be multiples of √10 or something like that? Wait, no, because if x and y are integers, then a and b might not necessarily be multiples of √10. Hmm, this is confusing.Wait, maybe I need to think differently. If I square both sides of the equation, I can get rid of the square roots. Let's try that. So, (√x + √y)^2 = (14√10)^2. Expanding the left side, I get x + 2√(xy) + y = 196*10, which is 1960. So, x + y + 2√(xy) = 1960.Hmm, okay, so x + y + 2√(xy) = 1960. Now, since x and y are integers, √(xy) must also be an integer because the other terms are integers. Let's denote √(xy) as c, where c is an integer. So, the equation becomes x + y + 2c = 1960.But I also know that c = √(xy), so c^2 = xy. So now I have two equations:1. x + y + 2c = 19602. xy = c^2This looks like a system of equations. Maybe I can express x and y in terms of c. Let's see.From the first equation, x + y = 1960 - 2c. From the second equation, xy = c^2. So, x and y are roots of the quadratic equation t^2 - (1960 - 2c)t + c^2 = 0.Hmm, let's write that down: t^2 - (1960 - 2c)t + c^2 = 0. To find integer solutions, the discriminant of this quadratic must be a perfect square. The discriminant D is [1960 - 2c]^2 - 4*1*c^2.Let me compute that: D = (1960 - 2c)^2 - 4c^2. Expanding (1960 - 2c)^2, I get 1960^2 - 2*1960*2c + (2c)^2, which is 1960^2 - 7840c + 4c^2. Then subtract 4c^2, so D = 1960^2 - 7840c + 4c^2 - 4c^2 = 1960^2 - 7840c.So, D = 1960^2 - 7840c. For the quadratic to have integer roots, D must be a perfect square. Let's denote D = k^2, where k is an integer. So, 1960^2 - 7840c = k^2.Hmm, this seems complicated. Maybe there's a better way to approach this problem. Let me think again.Going back to the original equation: √x + √y = 14√10. If I let √x = a√10 and √y = b√10, where a and b are integers, then the equation becomes a√10 + b√10 = 14√10. So, (a + b)√10 = 14√10, which implies a + b = 14.Okay, that seems promising. So, a + b = 14, where a and b are non-negative integers because they are coefficients of √10, which are square roots of integers. So, a and b can range from 0 to 14, as long as their sum is 14.Now, since √x = a√10, squaring both sides gives x = (a√10)^2 = a^2 * 10. Similarly, y = b^2 * 10. So, x and y must be multiples of 10, specifically 10 times a perfect square.So, for each pair (a, b) where a + b = 14, we get a corresponding pair (x, y) = (10a^2, 10b^2). Since a and b are non-negative integers, we need to count the number of such pairs.Let's list them out:- a = 0, b = 14: x = 0, y = 10*(14)^2 = 1960- a = 1, b = 13: x = 10*(1)^2 = 10, y = 10*(13)^2 = 1690- a = 2, b = 12: x = 10*(2)^2 = 40, y = 10*(12)^2 = 1440- a = 3, b = 11: x = 10*(3)^2 = 90, y = 10*(11)^2 = 1210- a = 4, b = 10: x = 10*(4)^2 = 160, y = 10*(10)^2 = 1000- a = 5, b = 9: x = 10*(5)^2 = 250, y = 10*(9)^2 = 810- a = 6, b = 8: x = 10*(6)^2 = 360, y = 10*(8)^2 = 640- a = 7, b = 7: x = 10*(7)^2 = 490, y = 10*(7)^2 = 490- a = 8, b = 6: x = 10*(8)^2 = 640, y = 10*(6)^2 = 360- a = 9, b = 5: x = 10*(9)^2 = 810, y = 10*(5)^2 = 250- a = 10, b = 4: x = 10*(10)^2 = 1000, y = 10*(4)^2 = 160- a = 11, b = 3: x = 10*(11)^2 = 1210, y = 10*(3)^2 = 90- a = 12, b = 2: x = 10*(12)^2 = 1440, y = 10*(2)^2 = 40- a = 13, b = 1: x = 10*(13)^2 = 1690, y = 10*(1)^2 = 10- a = 14, b = 0: x = 10*(14)^2 = 1960, y = 0So, that's 15 pairs in total. Each pair (a, b) gives a unique solution (x, y). Since a and b are non-negative integers, and we've considered all possible pairs where a + b = 14, we've covered all possible solutions.Wait a minute, but I need to make sure that x and y are non-negative integers as well. Looking at the solutions, x and y are all non-negative because a and b are non-negative, and squaring them and multiplying by 10 keeps them non-negative. So, all these solutions are valid.Therefore, the number of integer solutions (x, y) is 15.

💡Okay, so I have this problem where I need to convert a point from cylindrical coordinates to rectangular coordinates. The point given is (6, 5π/4, -3). Hmm, cylindrical coordinates... I remember that cylindrical coordinates are kind of like polar coordinates but with an extra z-component for height or depth. So, in cylindrical coordinates, we have (r, θ, z), where r is the radius, θ is the angle, and z is the height.Alright, I think the formula to convert from cylindrical to rectangular coordinates involves using trigonometry. Let me try to recall. I think it's something like x equals r times cosine of θ, and y equals r times sine of θ. And z remains the same, right? So, z is just z. That seems familiar.So, if I have (r, θ, z) = (6, 5π/4, -3), then I can plug these values into the formulas. Let me write that down:x = r * cos(θ) = 6 * cos(5π/4)y = r * sin(θ) = 6 * sin(5π/4)z = z = -3Okay, now I need to compute cos(5π/4) and sin(5π/4). I remember that 5π/4 is an angle in radians. Let me convert that to degrees to visualize it better. Since π radians is 180 degrees, π/4 is 45 degrees, so 5π/4 is 5 * 45 = 225 degrees. That's in the third quadrant, where both cosine and sine are negative.In the third quadrant, the reference angle is 225 - 180 = 45 degrees. So, cos(45°) is √2/2, and sin(45°) is also √2/2. But since we're in the third quadrant, both cosine and sine will be negative. So, cos(5π/4) = -√2/2 and sin(5π/4) = -√2/2.Alright, plugging these back into the equations for x and y:x = 6 * (-√2/2)y = 6 * (-√2/2)Let me compute these. 6 divided by 2 is 3, so:x = 6 * (-√2/2) = -3√2y = 6 * (-√2/2) = -3√2So, both x and y are -3√2. And z is just -3, as given.Wait, let me double-check. Is 5π/4 indeed in the third quadrant? Yes, because π is 180°, so 5π/4 is 225°, which is between 180° and 270°, so that's correct. And in the third quadrant, both x and y are negative, which matches our results.Also, checking the reference angle: 225° - 180° = 45°, so the reference angle is 45°, and cos and sin of 45° are both √2/2. Since we're in the third quadrant, both are negative. So, that seems right.So, putting it all together, the rectangular coordinates are (-3√2, -3√2, -3). Let me just write that in the required format.**Final Answer**The rectangular coordinates are boxed{(-3sqrt{2}, -3sqrt{2}, -3)}.

💡Okay, so I have this problem here: A car covers a distance of 642 kilometers in 6 and a half hours. I need to find its speed. Hmm, speed... I think speed is basically how fast something is going, right? So, if I remember correctly, speed is calculated by dividing the distance traveled by the time it took to travel that distance. Yeah, that sounds familiar from what I've learned before.Alright, so the distance is 642 kilometers. That seems straightforward. The time is 6 and a half hours. Wait, 6 and a half hours is the same as 6.5 hours, right? Yeah, because half an hour is 0.5 hours. So, I can write that as 6.5 hours to make the calculation easier.Now, to find the speed, I need to divide the distance by the time. So, that would be 642 kilometers divided by 6.5 hours. Let me write that down: 642 km ÷ 6.5 hours. Okay, so I need to do this division. Hmm, dividing by a decimal can be a bit tricky. Maybe I can convert 6.5 into a fraction to make it easier? 6.5 is the same as 13/2, right? So, dividing by 13/2 is the same as multiplying by 2/13. That might be easier.So, 642 km multiplied by 2/13 hours. Let me calculate that. First, multiply 642 by 2, which is 1284. Then, divide that by 13. Hmm, 13 times 98 is 1274, because 13 times 100 is 1300, and then subtract 13 times 2, which is 26, so 1300 minus 26 is 1274. Wait, but 1284 minus 1274 is 10. So, there's a remainder of 10. That means 1284 divided by 13 is 98 with a remainder of 10, or 98 and 10/13.So, the speed is 98 and 10/13 kilometers per hour. That's approximately 98.769 kilometers per hour. If I round that to the nearest whole number, it would be 99 kilometers per hour.Wait, let me double-check my calculations to make sure I didn't make any mistakes. Starting with 642 divided by 6.5. If I do 642 divided by 6, that's 107, and then I have to account for the extra half hour. So, 107 divided by 1.5? Wait, no, that doesn't seem right. Maybe I should approach it differently.Alternatively, I can multiply both the numerator and the denominator by 10 to eliminate the decimal. So, 642 times 10 is 6420, and 6.5 times 10 is 65. So now, it's 6420 divided by 65. Let's see, 65 times 98 is 6370, and 6420 minus 6370 is 50. So, 50 divided by 65 is approximately 0.769. So, altogether, that's 98.769 kilometers per hour, which again rounds to 99 kilometers per hour.Okay, so both methods give me the same result. That makes me more confident that 99 kilometers per hour is the correct speed of the car.Just to be thorough, let me think about units. The distance is in kilometers, and the time is in hours, so the speed should be in kilometers per hour, which makes sense. If I had different units, I might need to convert them, but in this case, everything is already in the right units.Also, intuitively, 642 kilometers in 6.5 hours seems like a pretty fast speed, but 99 kilometers per hour is reasonable for a car on a highway. So, that checks out.I think I've covered all the steps and double-checked my calculations. I'm pretty confident that the speed of the car is 99 kilometers per hour.

💡Okay, so I have this problem here: given that a and b are real numbers and they satisfy the equation a + 3b - 2 = 0, I need to find the minimum value of 2^a + 8^b. Hmm, let's see. I remember that when dealing with expressions like this, especially with exponents, sometimes you can use inequalities like AM-GM to find minima or maxima. But I'm not entirely sure how to apply it here. Let me think step by step.First, let's write down the given equation: a + 3b - 2 = 0. So, that simplifies to a + 3b = 2. Maybe I can express one variable in terms of the other to reduce the number of variables in the expression I need to minimize. Let me solve for a: a = 2 - 3b. Okay, so now I can substitute this into the expression 2^a + 8^b.Substituting a, we get 2^(2 - 3b) + 8^b. Hmm, 8 is 2^3, so 8^b is the same as (2^3)^b, which is 2^(3b). So, now the expression becomes 2^(2 - 3b) + 2^(3b). Let me write that down: 2^(2 - 3b) + 2^(3b). Now, let's see if I can simplify this further. 2^(2 - 3b) is equal to 2^2 * 2^(-3b), which is 4 * (1/2^(3b)). So, the expression becomes 4 * (1/2^(3b)) + 2^(3b). Let me denote x = 2^(3b). Then, the expression becomes 4*(1/x) + x. So, now I have an expression in terms of x: 4/x + x.Now, I need to find the minimum value of 4/x + x where x is a positive real number because 2^(3b) is always positive. This seems like a standard optimization problem. I think I can use calculus here, taking the derivative and setting it to zero to find the minimum. Alternatively, I remember that for expressions like x + c/x, the minimum occurs when x = sqrt(c). Let me check that.If I let f(x) = x + c/x, then the derivative f’(x) = 1 - c/x^2. Setting this equal to zero gives 1 - c/x^2 = 0, so x^2 = c, which means x = sqrt(c). So, in my case, c is 4, so x = sqrt(4) = 2. Therefore, the minimum occurs at x = 2. Plugging this back into f(x), we get 2 + 4/2 = 2 + 2 = 4. So, the minimum value is 4.Wait, but let me make sure I didn't skip any steps. Let me verify using calculus. Let f(x) = x + 4/x. Then f’(x) = 1 - 4/x^2. Setting f’(x) = 0, we have 1 - 4/x^2 = 0, so 4/x^2 = 1, which implies x^2 = 4, so x = 2 or x = -2. But since x = 2^(3b) is always positive, we discard x = -2. So, x = 2 is the critical point. Now, to ensure it's a minimum, we can check the second derivative. f''(x) = 8/x^3. At x = 2, f''(2) = 8/8 = 1, which is positive, so it's a minimum. Therefore, the minimum value is indeed 4.Alternatively, I could have used the AM-GM inequality from the start. AM-GM states that for non-negative real numbers, the arithmetic mean is greater than or equal to the geometric mean. So, for the two terms 2^(2 - 3b) and 2^(3b), their arithmetic mean is (2^(2 - 3b) + 2^(3b))/2, and their geometric mean is sqrt(2^(2 - 3b) * 2^(3b)) = sqrt(2^(2 - 3b + 3b)) = sqrt(2^2) = sqrt(4) = 2. Therefore, (2^(2 - 3b) + 2^(3b))/2 >= 2, which implies 2^(2 - 3b) + 2^(3b) >= 4. So, the minimum value is 4, achieved when the two terms are equal, i.e., when 2^(2 - 3b) = 2^(3b). Setting the exponents equal: 2 - 3b = 3b, so 2 = 6b, which gives b = 1/3. Then, substituting back into a + 3b = 2, we get a + 3*(1/3) = 2, so a + 1 = 2, hence a = 1. So, when a = 1 and b = 1/3, the expression 2^a + 8^b equals 2^1 + 8^(1/3) = 2 + 2 = 4, confirming the minimum value.Wait a second, let me double-check the substitution. 8^(1/3) is indeed 2 because 2^3 = 8, so 8^(1/3) = 2. And 2^1 is 2, so 2 + 2 is 4. Yep, that checks out.I think I've covered all the bases here. I used substitution to reduce the problem to a single variable, then applied both calculus and the AM-GM inequality to find the minimum. Both methods led me to the same conclusion, so I'm confident that the minimum value is 4, achieved when a = 1 and b = 1/3.