💡First, identify the given values: the radius ( r ) of the base is 2, and the height ( h ) of the cylinder is also 2.The formula for the lateral surface area of a cylinder is:[S = 2pi r h]Substitute the given values into the formula:[S = 2pi times 2 times 2 = 8pi]Therefore, the area of the unfolded lateral surface is ( 8pi ).

💡First, I recognize that the minor axis of the ellipse is equal to the diameter of the cylinder's base. Since the cylinder has a radius of 2, the diameter is (2 times 2 = 4).Next, the problem states that the major axis is 25% longer than the minor axis. To find the length of the major axis, I calculate 25% of the minor axis and add it to the minor axis length:[text{Major axis length} = 4 + (0.25 times 4) = 4 + 1 = 5.]Therefore, the length of the major axis is (5).

💡Okay, so I have this geometry problem here, and I need to figure out the measure of angle RPS. Let me try to visualize the problem first because understanding the diagram is crucial for solving geometry problems.The problem mentions points Q, R, P, and S arranged in a triangle, with QRS being a straight line. Hmm, so QRS is a straight line, which means that points Q, R, and S are colinear. That probably means that triangle PQS is formed with points P, Q, and S, and point R is somewhere on the line QRS.Given angles are:- Angle PQS = 55 degrees- Angle PSQ = 40 degrees- Angle QPR = 72 degreesAnd I need to find angle RPS.Alright, let me try to sketch this out mentally. Points Q, R, S are on a straight line. So, imagine a horizontal line with Q on the left, R in the middle, and S on the right. Point P is somewhere above this line, forming triangles PQS and possibly PRS.Given angle PQS is 55 degrees. That's the angle at point Q between lines QP and QS. Similarly, angle PSQ is 40 degrees, which is the angle at point S between lines SP and SQ. Angle QPR is 72 degrees, which is the angle at point P between lines PQ and PR.I think I need to find angle RPS, which is the angle at point P between lines PR and PS.Let me try to break this down step by step.First, in triangle PQS, I know two angles: angle PQS (55 degrees) and angle PSQ (40 degrees). Since the sum of angles in a triangle is 180 degrees, I can find the third angle at point P, which is angle QPS.So, angle QPS = 180 - 55 - 40 = 85 degrees.Okay, so angle QPS is 85 degrees. That's the angle at point P between lines PQ and PS.Now, I also know angle QPR is 72 degrees. That's the angle at point P between lines PQ and PR. So, if I imagine point R somewhere on the line QRS, then PR is another line from P to R.Since angle QPR is 72 degrees, and angle QPS is 85 degrees, I can think of angle RPS as the difference between these two angles.Wait, is that correct? Let me make sure.So, angle QPS is the angle between PQ and PS, which is 85 degrees. Angle QPR is the angle between PQ and PR, which is 72 degrees. So, if I subtract angle QPR from angle QPS, I should get angle RPS.So, angle RPS = angle QPS - angle QPR = 85 - 72 = 13 degrees.Hmm, that seems straightforward, but let me verify if there's another way to approach this to make sure I didn't miss anything.Alternatively, maybe I can use the Law of Sines or Law of Cosines in triangle PQS to find some side lengths, and then use those to find angle RPS. But since I don't have any side lengths given, that might not be necessary.Wait, but if I consider triangle PRS, I might need some more information. However, since QRS is a straight line, maybe triangles PQS and PRS are related somehow.Alternatively, perhaps I can consider the straight line QRS and use the concept of angles on a straight line summing up to 180 degrees. But I'm not sure how that would directly help here.Going back to my initial approach, I think it's correct because angle QPS is the larger angle at point P, and angle QPR is a part of it. So, subtracting them gives the remaining angle RPS.Let me double-check my calculations:Angle PQS = 55 degreesAngle PSQ = 40 degreesSo, angle QPS = 180 - 55 - 40 = 85 degreesAngle QPR = 72 degreesTherefore, angle RPS = 85 - 72 = 13 degreesYes, that seems consistent. I don't see any errors in this reasoning.I guess another way to think about it is that point R is on the line QRS, so PR is a line from P to some point R on QRS. The angle between PQ and PR is 72 degrees, and the angle between PQ and PS is 85 degrees, so the angle between PR and PS must be the difference, which is 13 degrees.I think that makes sense. I don't see any other angles or relationships that I'm missing here. So, I'm pretty confident that angle RPS is 13 degrees.

💡Okay, so I need to find the remainder when 3 raised to the power of 2048 is divided by 11. Hmm, that's a pretty big exponent. I remember that when dealing with remainders and exponents, modular arithmetic can be really helpful. Maybe I can use something like Fermat's Little Theorem here? Let me think.Fermat's Little Theorem says that if p is a prime number and a is not divisible by p, then a^(p-1) is congruent to 1 modulo p. Since 11 is a prime number and 3 is not divisible by 11, this theorem should apply. So, according to Fermat's Little Theorem, 3^(10) should be congruent to 1 mod 11. Let me check that.Calculating 3^10 directly would be tedious, but maybe I can find a pattern or break it down into smaller exponents. Let's compute the powers of 3 modulo 11 step by step:- 3^1 mod 11 = 3- 3^2 mod 11 = 9- 3^3 mod 11 = 27 mod 11 = 5 (since 27 - 2*11 = 5)- 3^4 mod 11 = 3^3 * 3 mod 11 = 5 * 3 = 15 mod 11 = 4- 3^5 mod 11 = 3^4 * 3 mod 11 = 4 * 3 = 12 mod 11 = 1Oh, interesting! So 3^5 is congruent to 1 mod 11. That means every multiple of 5 in the exponent will bring the result back to 1. So, if I have 3^5k, that will be congruent to 1^k = 1 mod 11.Now, the exponent I have is 2048. I need to express 2048 in terms of multiples of 5. Let's divide 2048 by 5 to see how many times 5 goes into it and what the remainder is.2048 divided by 5 is 409 with a remainder of 3, because 5*409 = 2045, and 2048 - 2045 = 3. So, 2048 = 5*409 + 3.Therefore, 3^2048 = 3^(5*409 + 3) = (3^5)^409 * 3^3.From earlier, we know that 3^5 ≡ 1 mod 11, so (3^5)^409 ≡ 1^409 ≡ 1 mod 11.Now, we just need to compute 3^3 mod 11, which we already found earlier to be 5.So, putting it all together: 3^2048 ≡ 1 * 5 ≡ 5 mod 11.Wait a second, let me double-check that. If 3^5 ≡ 1, then 3^(5k) ≡ 1^k ≡ 1. So, 3^(5*409) ≡ 1, and then multiplying by 3^3 gives 1*5 = 5. Yeah, that seems right.Alternatively, I could have used Fermat's Little Theorem directly. Since 3^10 ≡ 1 mod 11, I can write 2048 as 10*204 + 8. Then, 3^2048 = (3^10)^204 * 3^8. Since 3^10 ≡ 1, this simplifies to 1^204 * 3^8 ≡ 3^8 mod 11.Now, let's compute 3^8 mod 11. We already have 3^5 ≡ 1, so 3^8 = 3^(5+3) = 3^5 * 3^3 ≡ 1 * 5 ≡ 5 mod 11. So, both methods give the same result, which is reassuring.I think I'm confident now that the remainder when 3^2048 is divided by 11 is 5.

💡To solve for ( p ) in the equation ( 81^6 = 3^p ), I start by expressing 81 as a power of 3. Since ( 81 = 3^4 ), I can rewrite the equation as ( (3^4)^6 = 3^p ). Next, I apply the exponentiation rule ( (a^m)^n = a^{m times n} ) to simplify the left side of the equation. This gives me ( 3^{24} = 3^p ).Since the bases are the same, I can set the exponents equal to each other, resulting in ( 24 = p ). Therefore, the value of ( p ) is 24.

💡First, I'll define the variables to represent the ages of John, his mother, and his father.Let ( J ) be John's age, ( M ) be his mother's age, and ( F ) be his father's age.From the problem, I know three key pieces of information:1. John is half as old as his father, which means: [ F = 2J ]2. John's father is 4 years older than John's mother, so: [ F = M + 4 ]3. The age difference between John and his mother is 16 years, which gives: [ M = J + 16 ]Now, I'll substitute the expression for ( M ) from the third equation into the second equation to find ( F ) in terms of ( J ):[F = (J + 16) + 4 = J + 20]I now have two expressions for ( F ):[F = 2J quad text{and} quad F = J + 20]Setting these equal to each other to solve for ( J ):[2J = J + 20][2J - J = 20][J = 20]Finally, I'll substitute ( J = 20 ) back into the first equation to find ( F ):[F = 2 times 20 = 40]Therefore, John's father is 40 years old.