💡Okay, so I have this problem where I need to convert the general equation ( x^2 + y^2 - 4x = 0 ) into a parametric equation using ( theta ) as the parameter, given that ( y = 2costheta ). Hmm, let me think about how to approach this.First, I remember that parametric equations express ( x ) and ( y ) in terms of a third variable, which in this case is ( theta ). Since ( y ) is already given in terms of ( theta ), I just need to find an expression for ( x ) in terms of ( theta ) as well.Looking at the general equation ( x^2 + y^2 - 4x = 0 ), maybe I can rearrange it to make it easier to substitute ( y ). Let me try completing the square for the ( x ) terms. So, starting with:( x^2 - 4x + y^2 = 0 )To complete the square for ( x^2 - 4x ), I take half of the coefficient of ( x ), which is ( -4/2 = -2 ), and square it, getting ( (-2)^2 = 4 ). So, I add and subtract 4 to complete the square:( (x^2 - 4x + 4) - 4 + y^2 = 0 )This simplifies to:( (x - 2)^2 + y^2 = 4 )Okay, so now the equation is in the standard form of a circle with center at ( (2, 0) ) and radius 2. That makes sense because the original equation was quadratic in both ( x ) and ( y ).Since it's a circle, I can use trigonometric functions to parameterize ( x ) and ( y ). I know that for a circle centered at ( (h, k) ) with radius ( r ), the parametric equations are:( x = h + rcostheta )( y = k + rsintheta )But wait, in this case, the center is ( (2, 0) ) and the radius is 2. So, substituting these values in, I would get:( x = 2 + 2costheta )( y = 0 + 2sintheta )Hmm, but the problem gives ( y = 2costheta ). That's different from what I just wrote. In my parametrization, ( y ) is ( 2sintheta ), but the problem specifies ( y = 2costheta ). So, I need to adjust my parametrization to match this.Maybe I can use a different parameterization where ( y ) is expressed in terms of cosine instead of sine. Let me think. If I let ( y = 2costheta ), then to maintain the circle equation, ( x ) should be expressed using sine. Because in the standard circle parametrization, ( x ) and ( y ) are usually expressed with cosine and sine respectively, but they can be swapped as long as the relationship holds.So, if ( y = 2costheta ), then ( x ) should be ( 2 + 2sintheta ). Let me check if this satisfies the circle equation.Substituting ( x = 2 + 2sintheta ) and ( y = 2costheta ) into ( (x - 2)^2 + y^2 ):( (2 + 2sintheta - 2)^2 + (2costheta)^2 = (2sintheta)^2 + (2costheta)^2 = 4sin^2theta + 4cos^2theta = 4(sin^2theta + cos^2theta) = 4(1) = 4 )Yes, that works! So, the parametric equations are:( x = 2 + 2sintheta )( y = 2costheta )Let me just make sure I didn't make any mistakes. I started by completing the square to get the circle equation, recognized that I needed to parameterize it, adjusted for the given ( y = 2costheta ), and checked my work by plugging back into the circle equation. It seems correct.I think that's the solution. The parametric equations are ( x = 2 + 2sintheta ) and ( y = 2costheta ).

💡To find the next palindromic prime after 131, I need to consider the structure of three-digit palindromic numbers. A three-digit palindrome has the form ABA, where A is a digit from 1 to 9 and B is a digit from 0 to 9.Starting from 131, the next possible palindromic number would be 141. I need to check if 141 is a prime number. To do this, I can test divisibility by smaller prime numbers. 141 divided by 3 equals 47, which means 141 is divisible by 3 and therefore not a prime number.The next palindromic number is 151. I will check if 151 is a prime number by testing divisibility by primes less than its square root. 151 is not divisible by 2, 3, 5, 7, 11, or 13. Since there are no divisors other than 1 and itself, 151 is a prime number.Thus, the next palindromic prime after 131 is 151.

💡Alright, so I have this problem where I need to figure out how long it takes for a train to pass a bridge. The train is 250 meters long, the bridge is 150 meters long, and the train's speed is 35 kilometers per hour. Hmm, okay, let's break this down step by step.First, I think I need to understand what it means for the train to "pass" the bridge. Does that mean the entire train has to go over the bridge? So, not just the front of the train reaching the other end, but the very end of the train clearing the bridge completely? Yeah, that makes sense. So, the train has to cover its own length plus the length of the bridge.So, the total distance the train needs to travel to completely pass the bridge is the length of the train plus the length of the bridge. That would be 250 meters plus 150 meters, which equals 400 meters. Okay, got that part.Now, the speed of the train is given in kilometers per hour, but the distance is in meters. To make the units match, I think I need to convert the speed from km/h to meters per second (m/s). How do I do that? Well, I know that 1 kilometer is 1000 meters, and 1 hour is 3600 seconds. So, to convert km/h to m/s, I can multiply the speed by 1000 to get meters per hour and then divide by 3600 to get meters per second.Let me write that out: 35 km/h multiplied by 1000 meters/kilometer equals 35,000 meters per hour. Then, dividing by 3600 seconds per hour gives me approximately 9.7222 meters per second. Okay, so the train is moving at about 9.7222 m/s.Now, with the distance in meters and the speed in meters per second, I can use the formula for time, which is distance divided by speed. So, time equals 400 meters divided by 9.7222 meters per second. Let me do that calculation: 400 divided by 9.7222 is approximately 41.1528 seconds.Wait, does that make sense? Let me double-check. If the train is moving at almost 10 meters per second, then in 40 seconds, it would cover about 400 meters. Yeah, that seems reasonable. So, it would take roughly 41.15 seconds for the entire train to pass the bridge.I think I've got it. The total distance is 400 meters, the speed is about 9.7222 m/s, and dividing those gives me the time in seconds. Yeah, that seems to add up.

💡First, I need to determine the initial sum of money, which I'll denote as ( P ). The problem states that the amount after 3 years at simple interest is 920, and when the interest rate is increased by 3%, the amount becomes 992.I'll start by setting up the equation for the future value with the original interest rate ( r ):[920 = P + frac{P cdot r cdot 3}{100}]Next, I'll set up the equation with the increased interest rate ( r + 3 ):[992 = P + frac{P cdot (r + 3) cdot 3}{100}]To simplify, I'll multiply both equations by 100 to eliminate the denominator:[92000 = 100P + 3Pr][99200 = 100P + 3Pr + 9P]Subtracting the first equation from the second to eliminate ( P ) and solve for ( r ):[7200 = 9P]Finally, solving for ( P ):[P = frac{7200}{9} = 800]Therefore, the initial sum of money is 800.

💡First, I need to determine the elements of set ( B ) by calculating all possible differences ( x - y ) where ( x ) and ( y ) are elements of set ( A = {0, 1, 2} ).I'll start by considering each value of ( x ) in set ( A ) and subtracting each value of ( y ) in set ( A ):1. When ( x = 0 ): - ( 0 - 0 = 0 ) - ( 0 - 1 = -1 ) - ( 0 - 2 = -2 )2. When ( x = 1 ): - ( 1 - 0 = 1 ) - ( 1 - 1 = 0 ) - ( 1 - 2 = -1 )3. When ( x = 2 ): - ( 2 - 0 = 2 ) - ( 2 - 1 = 1 ) - ( 2 - 2 = 0 )Next, I'll compile all these results into set ( B ), ensuring that each element is unique:[ B = {-2, -1, 0, 1, 2} ]Finally, I'll count the number of distinct elements in set ( B ):There are 5 distinct elements in set ( B ).Therefore, the number of elements in set ( B ) is 5.

💡First, I need to determine the number of edges in a triangle. A triangle is a three-sided polygon, so it has 3 edges.Next, I'll find the number of edges in a rectangle. A rectangle is a four-sided polygon, so it has 4 edges.Finally, I'll add the number of edges from both shapes together to find the total number of edges. Adding 3 edges from the triangle to 4 edges from the rectangle gives a total of 7 edges.