💡Okay, so I have this problem with two intersecting circles, O and C. Circle O is given by the equation x² + y² = 25, which I recognize as a circle centered at the origin (0,0) with a radius of 5 because 25 is 5 squared. Circle C has the equation x² + y² - 4x - 2y - 20 = 0. Hmm, that looks a bit more complicated. I think I need to rewrite it in the standard form of a circle to find its center and radius.Let me try completing the squares for the x and y terms in circle C's equation. So, starting with x² - 4x and y² - 2y. For the x terms, I take the coefficient of x, which is -4, divide by 2 to get -2, and then square it to get 4. Similarly, for the y terms, the coefficient is -2, divide by 2 to get -1, and square it to get 1. So, I'll add and subtract these squares to complete the squares.Rewriting circle C's equation:x² - 4x + 4 - 4 + y² - 2y + 1 - 1 - 20 = 0This simplifies to:(x - 2)² + (y - 1)² - 4 - 1 - 20 = 0Which is:(x - 2)² + (y - 1)² - 25 = 0So, moving the constant to the other side:(x - 2)² + (y - 1)² = 25Wait, that's interesting. So circle C is also a circle with radius 5, centered at (2,1). So both circles have the same radius, 5, but different centers: circle O at (0,0) and circle C at (2,1). That means they are congruent circles intersecting each other.Now, the problem is asking for the length of the common chord AB where they intersect. I remember that the common chord is the line segment connecting the two intersection points A and B. To find its length, I think I need to find the distance between points A and B.But how do I find points A and B? Maybe I can solve the two equations simultaneously. Let me write down both equations:Circle O: x² + y² = 25Circle C: (x - 2)² + (y - 1)² = 25If I expand circle C's equation:x² - 4x + 4 + y² - 2y + 1 = 25Simplify:x² + y² - 4x - 2y + 5 = 25Subtract 25 from both sides:x² + y² - 4x - 2y - 20 = 0Wait, that's the original equation given for circle C. So, to find the points of intersection, I can subtract the equation of circle O from circle C. Let's do that:(x² + y² - 4x - 2y - 20) - (x² + y² - 25) = 0Simplify:-4x - 2y - 20 + 25 = 0Which becomes:-4x - 2y + 5 = 0Divide both sides by -1:4x + 2y - 5 = 0So, the equation 4x + 2y - 5 = 0 is the equation of the line AB, which is the common chord. Now, to find the length of AB, I can find the distance between points A and B on this line, but I don't know their exact coordinates yet.Alternatively, I remember that the length of the common chord can be found using the distance between the centers of the two circles and their radii. Since both circles have the same radius, 5, and their centers are at (0,0) and (2,1), I can calculate the distance between the centers first.The distance between centers, let's call it d, can be found using the distance formula:d = sqrt[(2 - 0)² + (1 - 0)²] = sqrt[4 + 1] = sqrt[5]So, the distance between the centers is sqrt(5). Now, since both circles have the same radius, the triangle formed by the centers and one of the intersection points is an isosceles triangle with sides 5, 5, and sqrt(5). The common chord AB is the base of this isosceles triangle.I recall that in such a triangle, the length of the base can be found using the formula:Length of AB = 2 * sqrt(r² - (d/2)²)Where r is the radius, and d is the distance between the centers.Plugging in the values:Length of AB = 2 * sqrt(5² - (sqrt(5)/2)²)Calculate inside the square root:5² = 25(sqrt(5)/2)² = 5/4So, 25 - 5/4 = (100/4 - 5/4) = 95/4Therefore, Length of AB = 2 * sqrt(95/4) = 2 * (sqrt(95)/2) = sqrt(95)Wait, that seems a bit complicated. Let me double-check my steps.1. Both circles have radius 5.2. Distance between centers is sqrt(5).3. The formula for the length of the common chord when two circles have equal radii is 2 * sqrt(r² - (d/2)²).Yes, that seems correct. So, plugging in:2 * sqrt(25 - (sqrt(5)/2)²) = 2 * sqrt(25 - 5/4) = 2 * sqrt(95/4) = sqrt(95)Alternatively, I could have found the points of intersection by solving the two circle equations. Let me try that method to verify.From the line equation 4x + 2y - 5 = 0, I can express y in terms of x:2y = -4x + 5y = (-4x + 5)/2Now, substitute this into circle O's equation:x² + [(-4x + 5)/2]^2 = 25Calculate [(-4x + 5)/2]^2:= (16x² - 40x + 25)/4So, the equation becomes:x² + (16x² - 40x + 25)/4 = 25Multiply both sides by 4 to eliminate the denominator:4x² + 16x² - 40x + 25 = 100Combine like terms:20x² - 40x + 25 - 100 = 0Simplify:20x² - 40x - 75 = 0Divide all terms by 5:4x² - 8x - 15 = 0Now, solve this quadratic equation for x:Using the quadratic formula, x = [8 ± sqrt(64 + 240)] / 8Because the equation is 4x² - 8x - 15 = 0, so a=4, b=-8, c=-15Discriminant D = b² - 4ac = 64 - 4*4*(-15) = 64 + 240 = 304So, x = [8 ± sqrt(304)] / 8Simplify sqrt(304): sqrt(16*19) = 4*sqrt(19)Thus, x = [8 ± 4sqrt(19)] / 8 = [2 ± sqrt(19)] / 2So, x = 1 + (sqrt(19)/2) and x = 1 - (sqrt(19)/2)Now, find the corresponding y values using y = (-4x + 5)/2For x = 1 + (sqrt(19)/2):y = (-4*(1 + sqrt(19)/2) + 5)/2= (-4 - 2sqrt(19) + 5)/2= (1 - 2sqrt(19))/2= 1/2 - sqrt(19)For x = 1 - (sqrt(19)/2):y = (-4*(1 - sqrt(19)/2) + 5)/2= (-4 + 2sqrt(19) + 5)/2= (1 + 2sqrt(19))/2= 1/2 + sqrt(19)So, the points of intersection A and B are:A: (1 + sqrt(19)/2, 1/2 - sqrt(19))B: (1 - sqrt(19)/2, 1/2 + sqrt(19))Now, to find the distance between A and B, use the distance formula:Distance AB = sqrt[(x2 - x1)² + (y2 - y1)²]Calculate x2 - x1:[1 - sqrt(19)/2] - [1 + sqrt(19)/2] = -sqrt(19)Calculate y2 - y1:[1/2 + sqrt(19)] - [1/2 - sqrt(19)] = 2sqrt(19)So, Distance AB = sqrt[(-sqrt(19))² + (2sqrt(19))²] = sqrt[19 + 4*19] = sqrt[19 + 76] = sqrt[95]Yes, that matches the previous result. So, the length of the common chord AB is sqrt(95).I think that's correct. Both methods led me to the same answer, so I'm confident that the length of AB is sqrt(95).

💡Okay, so I've got this problem here about quadratic equations, and I need to figure out the correct equation based on the mistakes two students made. Let me try to break this down step by step.First, I remember that a quadratic equation can be written in the form ( x^2 + bx + c = 0 ), where ( b ) is the coefficient of the ( x ) term and ( c ) is the constant term. The roots of this equation can be found using the quadratic formula, but I also recall Vieta's formulas, which relate the coefficients of the polynomial to the sum and product of its roots. Specifically, for a quadratic equation ( x^2 + bx + c = 0 ), the sum of the roots ( alpha + beta ) is equal to ( -b ), and the product ( alpha beta ) is equal to ( c ).Now, let's look at the problem. The first student incorrectly figured out only the constant term and ended up with roots 7 and 3. The second student incorrectly calculated only the coefficient of the ( x ) term and derived roots -12 and 3. So, each student made a mistake in one part of the equation, but got the other part right.Starting with the first student: they got the constant term wrong but the coefficient of ( x ) right. If their roots are 7 and 3, then using Vieta's formulas, the sum of the roots is ( 7 + 3 = 10 ). Since the sum of the roots is equal to ( -b ), this means that ( -b = 10 ), so ( b = -10 ). So, the coefficient of ( x ) in the correct equation should be -10.Moving on to the second student: they got the coefficient of ( x ) wrong but the constant term right. Their roots are -12 and 3. Again, using Vieta's formulas, the product of the roots is ( (-12) times 3 = -36 ). Since the product of the roots is equal to ( c ), this means that ( c = -36 ). So, the constant term in the correct equation should be -36.Putting it all together, the correct quadratic equation should have ( b = -10 ) and ( c = -36 ). Therefore, the equation is ( x^2 - 10x - 36 = 0 ).Let me double-check to make sure I didn't mix up anything. The first student had the right coefficient for ( x ) but the wrong constant term. Their roots gave us the correct sum, which translates to the correct ( b ). The second student had the right constant term but the wrong coefficient for ( x ). Their roots gave us the correct product, which translates to the correct ( c ). So, combining the correct ( b ) and ( c ), I get the equation ( x^2 - 10x - 36 = 0 ).Looking at the answer choices, option E is ( x^2 - 10x - 36 = 0 ), which matches what I derived. So, I think that's the correct answer.**Final Answer**boxed{E}

💡First, I need to translate the problem into a mathematical inequality. The phrase "the product of 5 and a number" can be written as (5n), where (n) is the unknown number. The expression "the difference when this product is subtracted from 25" translates to (25 - 5n). The problem states that this difference is greater than 15, so the inequality becomes:[25 - 5n > 15]Next, I'll solve this inequality step by step. Starting with the original inequality:[25 - 5n > 15]I'll subtract 15 from both sides to simplify:[10 - 5n > 0]Then, I'll add (5n) to both sides to isolate the term with (n):[10 > 5n]Finally, I'll divide both sides by 5 to solve for (n):[2 > n]This means that (n) must be less than 2. Since we're looking for positive integers, the only value that satisfies this condition is (n = 1).Therefore, there is only one positive integer that satisfies the given conditions.

💡First, I need to determine the total cost of the meal before the tip by adding the costs of the appetizer, entrees, and dessert.Next, I'll calculate the tip amount by subtracting the total cost before the tip from the entire price of the meal.Finally, to find the tip percentage, I'll divide the tip amount by the total cost before the tip and then multiply by 100 to convert it to a percentage.

💡Okay, so I have this problem here about a set M, which is defined as all real numbers x such that -2 is less than x, and x is less than 3. So, M is the interval from -2 to 3, not including the endpoints. The question is asking which of the statements A, B, C, or D is correct.Let me go through each option one by one.Starting with option A: 2.5 is an element of M. Hmm, 2.5 is a number between -2 and 3. Since M is defined as all numbers greater than -2 and less than 3, 2.5 definitely fits into that range. So, 2.5 is in M. That seems correct.Moving on to option B: 0 is a subset of M. Wait, hold on. I think I might be mixing up some concepts here. When we say something is a subset, we're talking about one set being entirely contained within another set. But 0 is just a single number, not a set. So, saying 0 is a subset of M doesn't quite make sense. Instead, if we wanted to talk about 0 in relation to M, we should say 0 is an element of M, which it is, because 0 is between -2 and 3. So, option B is incorrect because it's using the wrong notation. It should be 0 ∈ M, not 0 ⊆ M.Next is option C: The empty set is an element of M. Again, this seems a bit confusing. The empty set, denoted by ∅, is a set that contains no elements. Now, in set theory, the empty set is a subset of every set, which means ∅ ⊆ M. But the question here is asking if ∅ is an element of M, which would mean that ∅ is one of the elements inside M. However, M is a set of real numbers, and the empty set is not a real number. Therefore, ∅ is not an element of M. So, option C is incorrect.Finally, option D: The set M is finite. Well, M is the interval from -2 to 3, which includes infinitely many real numbers. Between any two real numbers, there are infinitely many others. So, M is definitely an infinite set, not a finite one. Therefore, option D is incorrect.So, after going through each option, the only correct statement is option A: 2.5 is an element of M.

💡Alright, so I've got this problem here about roller skates and wheels. Let's see, it says that on the last night the roller skate rink was open, 40 people showed up to roller skate one last time. And the question is, when all 40 people skated at one time, how many wheels were on the floor?Okay, first off, I need to figure out how many wheels each person has. I know that roller skates typically have wheels, right? So, if someone is roller skating, they're probably wearing roller skates on both feet. Now, how many wheels does a roller skate have? I think it's usually four wheels per skate. So, if each foot has a roller skate with four wheels, then per person, that would be 4 wheels on each foot, times two feet. So, 4 times 2 equals 8 wheels per person.Wait, let me double-check that. Sometimes roller skates can have more wheels, like some have five or even more, but I think the standard is four wheels per skate. Yeah, I'll stick with four wheels per skate for this problem.So, if each person has 8 wheels, and there are 40 people, then the total number of wheels should be 8 wheels per person times 40 people. That would be 8 times 40, which is 320 wheels.But hold on, is there any chance that some people might not have both skates on? Like, maybe someone is just wearing one roller skate and one regular shoe? The problem doesn't specify, so I think it's safe to assume that everyone is fully roller skating, meaning both feet are on roller skates.Also, are there any wheels on the floor besides the ones on the roller skates? Well, roller skates themselves are on the floor, so all the wheels are on the floor as well. So, there's no need to consider any other wheels, like maybe from a skateboard or something else.So, to recap: 40 people, each with two roller skates, each roller skate having four wheels. So, 40 times 2 is 80 roller skates, and 80 times 4 is 320 wheels. Yeah, that seems right.I think I've got it. The total number of wheels on the floor when all 40 people are skating at the same time is 320 wheels.