💡First, let's define the number of each type of coin. Let ( p ) represent the number of pennies. According to the problem, there are three times as many dimes as pennies, so the number of dimes is ( 3p ). Additionally, there are twice as many quarters as dimes, which means the number of quarters is ( 2 times 3p = 6p ).Next, we calculate the total value contributed by each type of coin:- The value of pennies is ( 0.01p ) dollars.- The value of dimes is ( 0.10 times 3p = 0.30p ) dollars.- The value of quarters is ( 0.25 times 6p = 1.50p ) dollars.Adding these values together gives the total amount of money in the container:[text{Total value} = 0.01p + 0.30p + 1.50p = 1.81p text{ dollars}]To determine which of the given options could be the total amount, we convert the total value into cents:[1.81p times 100 = 181p text{ cents}]We then check which of the provided choices, when multiplied by 100, is a multiple of 181:- ( 270 times 100 = 27000 ) (not a multiple of 181)- ( 285 times 100 = 28500 ) (not a multiple of 181)- ( 297 times 100 = 29700 ) (not a multiple of 181)- ( 315 times 100 = 31500 ) (a multiple of 181, since ( 31500 div 181 = 174 ))- ( 330 times 100 = 33000 ) (not a multiple of 181)The only choice that results in a multiple of 181 is 31500 cents, which corresponds to 315 dollars.

💡Okay, so I have this problem where I need to figure out the relationship between three values: a, b, and c. They’re defined as follows:- a is the logarithm base 2 of 0.3, so a = log₂(0.3)- b is 2 raised to the power of 0.3, so b = 2^0.3- c is 0.3 raised to the power of 0.2, so c = 0.3^0.2I need to determine how these three values compare to each other—like which one is the smallest, which is in the middle, and which is the largest. Alright, let me start by recalling some basic properties of logarithms and exponents because I think that will help me here. First, let's tackle a = log₂(0.3). I know that logarithms can be tricky, especially when dealing with numbers less than 1. Remember that log₂(1) is 0 because 2^0 = 1. So, if I have a number less than 1, like 0.3, its logarithm base 2 should be negative because the logarithm of a number less than 1 is negative. That makes sense because 2 raised to a negative power gives a fraction. So, a is definitely less than 0.Next, let's look at b = 2^0.3. Since 0.3 is a positive exponent, and 2 is a number greater than 1, 2 raised to any positive power will be greater than 1. For example, 2^0 = 1, and as the exponent increases, the result increases. So, 2^0.3 must be greater than 1. That means b is greater than 1.Now, c = 0.3^0.2. This is a bit more confusing because both the base and the exponent are less than 1. I remember that when you raise a number between 0 and 1 to a positive exponent, the result is still between 0 and 1, but it's actually larger than the original number. For example, 0.5^2 = 0.25, which is smaller, but 0.5^0.5 is approximately 0.707, which is larger than 0.5. Wait, so in this case, 0.3^0.2 should be greater than 0.3 but less than 1 because the exponent is between 0 and 1. So, c is between 0 and 1.Putting it all together:- a is less than 0- c is between 0 and 1- b is greater than 1So, in terms of order from smallest to largest, it should be a < c < b.But let me double-check to make sure I didn't make any mistakes.Starting with a = log₂(0.3). Since 0.3 is less than 1, log₂(0.3) is negative. To get a better sense of how negative it is, I can approximate it. I know that log₂(1/2) = -1, and 0.3 is less than 1/2, so log₂(0.3) should be less than -1. Let me calculate it roughly. Using the change of base formula: log₂(0.3) = ln(0.3)/ln(2). ln(0.3) is approximately -1.20397, and ln(2) is approximately 0.6931. So, log₂(0.3) ≈ -1.20397 / 0.6931 ≈ -1.737. So, a is approximately -1.737.Next, b = 2^0.3. Let's approximate this. I know that 2^0.3 is the same as e^(0.3 * ln 2). ln 2 is approximately 0.6931, so 0.3 * 0.6931 ≈ 0.2079. Then, e^0.2079 is approximately 1.230. So, b is approximately 1.230.Now, c = 0.3^0.2. Let's calculate this. Again, using natural logarithms: 0.3^0.2 = e^(0.2 * ln 0.3). ln 0.3 is approximately -1.20397, so 0.2 * (-1.20397) ≈ -0.2408. Then, e^-0.2408 is approximately 0.7866. So, c is approximately 0.7866.So, plugging in the approximate values:- a ≈ -1.737- c ≈ 0.7866- b ≈ 1.230So, ordering them from smallest to largest: a < c < b.Wait, but I just thought of something. When dealing with exponents, sometimes the behavior can be counterintuitive. For example, 0.3^0.2 is the same as (3/10)^(1/5). So, taking the fifth root of 3/10. Since 3/10 is 0.3, and the fifth root of 0.3 is a number slightly less than 1, but still positive. So, that aligns with my earlier conclusion that c is between 0 and 1.Another way to think about c is that since 0.3 is less than 1, raising it to a power less than 1 will actually make it larger, but it's still less than 1. So, 0.3^0.2 is greater than 0.3 but less than 1.So, putting it all together, a is negative, c is between 0 and 1, and b is greater than 1. Therefore, the order is a < c < b.I think that makes sense. I don't see any mistakes in my reasoning, and the approximations back it up. So, I feel confident that the relationship is a < c < b.

💡Alright, so I have this triangle problem here, and I need to find angle B and the area of triangle ABC. Let me see what information I've been given:1. The equation: 2b cos A + a = 2c2. c = 83. sin A = (3√3)/14Hmm, okay. So, I need to find angle B and the area. Let me break this down step by step.First, I know that in any triangle, the sum of the angles is π radians (or 180 degrees). So, A + B + C = π. That might come in handy later.I also remember the Law of Sines, which says that a/sin A = b/sin B = c/sin C = 2R, where R is the radius of the circumscribed circle. Maybe that can help me relate the sides and angles.Given that sin A is provided, I can find cos A using the Pythagorean identity: sin² A + cos² A = 1. Let me calculate that first.sin A = (3√3)/14, so sin² A = (9*3)/196 = 27/196.Therefore, cos² A = 1 - 27/196 = (196 - 27)/196 = 169/196.So, cos A = √(169/196) = 13/14. Since angle A is in a triangle, it must be between 0 and π, so cos A is positive, meaning angle A is acute.Alright, so cos A = 13/14.Now, looking back at the given equation: 2b cos A + a = 2c.We know c = 8, so 2c = 16.So, 2b cos A + a = 16.Let me write that as: 2b*(13/14) + a = 16.Simplify 2b*(13/14): that's (26/14)b = (13/7)b.So, (13/7)b + a = 16.Hmm, okay. So, I have an equation involving sides a and b.I need another equation to relate a and b. That's where the Law of Sines comes in.From the Law of Sines: a/sin A = b/sin B = c/sin C.We know c = 8, and sin A = (3√3)/14. So, let's write:a = (sin A / sin C) * cWait, no, actually, a/sin A = c/sin C, so a = (c * sin A)/sin C.Similarly, b = (c * sin B)/sin C.But I don't know sin C yet. Hmm.Wait, since A + B + C = π, then C = π - A - B.So, sin C = sin(π - A - B) = sin(A + B).Using the sine addition formula: sin(A + B) = sin A cos B + cos A sin B.So, sin C = sin A cos B + cos A sin B.Therefore, a = (c * sin A)/sin C = (8 * (3√3)/14) / (sin A cos B + cos A sin B).Similarly, b = (8 * sin B)/sin C = (8 * sin B)/(sin A cos B + cos A sin B).Hmm, this is getting a bit complicated. Maybe I can substitute these expressions for a and b into the equation 2b cos A + a = 16.Let me try that.First, let's write a and b in terms of sin B and cos B.From above, a = (8 * (3√3)/14) / (sin A cos B + cos A sin B).Similarly, b = (8 * sin B) / (sin A cos B + cos A sin B).Let me denote sin A as sA and cos A as cA for simplicity.So, sA = (3√3)/14, cA = 13/14.So, a = (8 * sA) / (sA cos B + cA sin B) = (8 * (3√3)/14) / ((3√3)/14 cos B + (13/14) sin B).Similarly, b = (8 sin B) / ((3√3)/14 cos B + (13/14) sin B).So, now, plug these into the equation: 2b cos A + a = 16.So, 2 * [ (8 sin B) / ((3√3)/14 cos B + (13/14) sin B) ] * (13/14) + [ (8 * (3√3)/14) / ((3√3)/14 cos B + (13/14) sin B) ] = 16.Wow, that's a mouthful. Let me try to simplify this step by step.First, let's compute 2b cos A:2b cos A = 2 * [ (8 sin B) / ((3√3)/14 cos B + (13/14) sin B) ] * (13/14)= 2 * (8 sin B) * (13/14) / [ (3√3)/14 cos B + (13/14) sin B ]= (16 sin B * 13/14) / [ (3√3 cos B + 13 sin B)/14 ]= (16 sin B * 13/14) * (14 / (3√3 cos B + 13 sin B))= (16 sin B * 13) / (3√3 cos B + 13 sin B )Similarly, a = (8 * (3√3)/14) / [ (3√3)/14 cos B + (13/14) sin B ]= (24√3 /14 ) / [ (3√3 cos B + 13 sin B)/14 ]= (24√3 /14 ) * (14 / (3√3 cos B + 13 sin B))= 24√3 / (3√3 cos B + 13 sin B )So, now, the equation becomes:[ (16 sin B * 13) / (3√3 cos B + 13 sin B ) ] + [ 24√3 / (3√3 cos B + 13 sin B ) ] = 16Since both terms have the same denominator, we can combine them:[ (16 sin B * 13) + 24√3 ] / (3√3 cos B + 13 sin B ) = 16Multiply both sides by the denominator:16 sin B * 13 + 24√3 = 16 * (3√3 cos B + 13 sin B )Let me compute each part:Left side: 16*13 sin B + 24√3 = 208 sin B + 24√3Right side: 16*3√3 cos B + 16*13 sin B = 48√3 cos B + 208 sin BSo, the equation becomes:208 sin B + 24√3 = 48√3 cos B + 208 sin BSubtract 208 sin B from both sides:24√3 = 48√3 cos BDivide both sides by 48√3:24√3 / (48√3) = cos BSimplify:24/48 = cos BSo, 1/2 = cos BTherefore, cos B = 1/2So, angle B is arccos(1/2). The solutions for cos B = 1/2 in the range [0, π] are B = π/3 (60 degrees) and B = 5π/3, but 5π/3 is more than π, so it's not possible in a triangle. So, B = π/3.Alright, so part (I) is solved: angle B is π/3 radians or 60 degrees.Now, moving on to part (II): finding the area of triangle ABC.I know that the area of a triangle can be found using the formula:Area = (1/2)ab sin CBut I don't know sides a and b yet, but maybe I can find them.Alternatively, since I know angle B and sides related to it, maybe I can use another formula.Wait, I know side c = 8, and I know angles A and B, so I can find angle C.Since A + B + C = π, so C = π - A - B.I know sin A, which is (3√3)/14, and angle B is π/3, so sin B = sin(π/3) = √3/2.But I need angle C. Let me compute angle C.First, I need angle A. Since sin A = (3√3)/14, I can find angle A.Let me compute angle A:sin A = (3√3)/14 ≈ (5.196)/14 ≈ 0.371So, angle A ≈ arcsin(0.371) ≈ 21.7 degrees.But since sin A is positive and less than 1, angle A is acute, so that's the only solution.So, angle C = 180 - 21.7 - 60 ≈ 98.3 degrees.But let me do this more accurately.Alternatively, since I know sin A and cos A, I can find angle A using arctangent.tan A = sin A / cos A = (3√3/14) / (13/14) = 3√3 /13 ≈ 5.196 /13 ≈ 0.3997So, angle A ≈ arctan(0.3997) ≈ 21.7 degrees.So, angle C ≈ 180 - 21.7 - 60 ≈ 98.3 degrees.But maybe I can find sin C using the sine addition formula.Since C = π - A - B, sin C = sin(A + B).Wait, no, sin(π - x) = sin x, so sin C = sin(A + B).But A + B = π - C, so sin C = sin(A + B).Wait, actually, sin(π - x) = sin x, so sin C = sin(A + B).But since A + B = π - C, sin(A + B) = sin(π - C) = sin C.Wait, that's circular. Maybe I should use the sine addition formula.So, sin C = sin(A + B) = sin A cos B + cos A sin B.We know sin A, cos A, sin B, and cos B.So, sin C = (3√3/14)(1/2) + (13/14)(√3/2)= (3√3/28) + (13√3/28)= (16√3)/28Simplify: 16/28 = 4/7, so sin C = (4√3)/7.Okay, so sin C = 4√3 /7.Now, using the Law of Sines, a/sin A = c/sin C.So, a = c * (sin A / sin C) = 8 * ( (3√3/14) / (4√3/7) )Simplify:First, (3√3/14) / (4√3/7) = (3√3/14) * (7/(4√3)) = (3*7)/(14*4) * (√3/√3) = (21)/(56) * 1 = 3/8.So, a = 8 * (3/8) = 3.Similarly, b = c * (sin B / sin C) = 8 * ( (√3/2) / (4√3/7) )Simplify:(√3/2) / (4√3/7) = (√3/2) * (7/(4√3)) = (7)/(8) * (√3/√3) = 7/8.So, b = 8 * (7/8) = 7.So, sides a = 3, b = 7, c = 8.Now, to find the area, I can use Heron's formula.First, compute the semi-perimeter, s = (a + b + c)/2 = (3 + 7 + 8)/2 = 18/2 = 9.Then, area = √[s(s - a)(s - b)(s - c)] = √[9*(9 - 3)*(9 - 7)*(9 - 8)] = √[9*6*2*1] = √[108] = √[36*3] = 6√3.Alternatively, I could use the formula (1/2)ab sin C.Using a = 3, b = 7, and sin C = 4√3/7.Area = (1/2)*3*7*(4√3/7) = (1/2)*3*4√3 = (1/2)*12√3 = 6√3.Either way, the area is 6√3.So, summarizing:(I) Angle B is π/3 radians or 60 degrees.(II) The area of triangle ABC is 6√3.**Final Answer**Ⅰ. boxed{dfrac{pi}{3}}Ⅱ. boxed{6sqrt{3}}

💡Okay, so I have this problem where I'm given a random variable X that follows a binomial distribution with parameters n=10 and p=0.6. Then there's another random variable η defined as η = 8 - 2X. I need to find the variance of η, which is denoted as Dη.First, I remember that for a binomial distribution B(n, p), the variance of X, which is DX, can be calculated using the formula:DX = n * p * (1 - p)So, plugging in the values, n is 10 and p is 0.6. Let me compute that:DX = 10 * 0.6 * (1 - 0.6) = 10 * 0.6 * 0.4Hmm, 10 times 0.6 is 6, and 6 times 0.4 is 2.4. So, DX is 2.4.Now, I need to find the variance of η, which is 8 - 2X. I recall that variance has a property where if you have a linear transformation of a random variable, like aX + b, the variance is a squared times the variance of X. The constant term b doesn't affect the variance because variance measures spread, and adding a constant doesn't change the spread.So, in this case, η is 8 - 2X, which can be written as -2X + 8. Here, a is -2 and b is 8. Therefore, the variance of η should be (-2)^2 times DX.Let me calculate that:Dη = (-2)^2 * DX = 4 * 2.44 times 2.4 is... let me see, 4 times 2 is 8, and 4 times 0.4 is 1.6, so total is 9.6.So, the variance of η is 9.6.Wait, just to make sure I didn't make a mistake. Let me go through the steps again.1. X ~ B(10, 0.6), so DX = 10 * 0.6 * 0.4 = 2.4. That seems right.2. η = 8 - 2X. So, variance of η is variance of (-2X + 8). Since variance is unaffected by the constant term, it's just variance of -2X.3. Variance of aX is a^2 times variance of X. So, (-2)^2 = 4, and 4 * 2.4 = 9.6.Yes, that seems consistent. I don't think I made any calculation errors here.Just to double-check, maybe I can think about it differently. If X is binomial, then it's the number of successes in 10 trials with probability 0.6 each. The variance being 2.4 makes sense because it's not too high or too low for these parameters.Then, scaling X by -2 would stretch the variance by a factor of 4, which is why it's 9.6. Adding 8 doesn't change anything about the spread, so the variance remains 9.6.I think that's solid. I don't see any issues with the reasoning or calculations.**Final Answer**The variance of η is boxed{9.6}.

💡First, I need to determine the relative speed between the train and the motorbike. Since both are moving in the same direction, the relative speed is the difference between their speeds.Next, I'll convert the relative speed from kilometers per hour to meters per second to match the units of the train's length.Finally, I'll calculate the time it takes for the train to overtake the motorbike by dividing the length of the train by the relative speed in meters per second.

💡Okay, so I have this function here: y equals the square root of (x plus 1) divided by x. I need to figure out the range of the independent variable x. Hmm, let's see. I remember that when dealing with functions, especially ones with square roots and denominators, there are certain restrictions on the domain, which is the set of all possible x-values that make the function defined.First, let's look at the square root part: sqrt(x + 1). For a square root to be defined in real numbers, the expression inside the square root, which is x + 1 in this case, must be greater than or equal to zero. So, I can write that as:x + 1 ≥ 0Solving this inequality, I subtract 1 from both sides:x ≥ -1Okay, so x has to be greater than or equal to -1. That's one condition.Next, let's look at the denominator: x. We can't have division by zero, so x cannot be zero. That gives me another condition:x ≠ 0So, putting these two conditions together, x has to be greater than or equal to -1, but it can't be zero. So, x is in the interval from -1 to infinity, excluding zero.Wait, let me make sure I'm not missing anything here. Is there any other restriction? The numerator is a square root, which we've already handled, and the denominator is just x, so as long as x isn't zero, we're good. I don't think there are any other restrictions because the square root doesn't impose any additional constraints beyond what's already been considered.Let me visualize this on a number line. Starting from -1, we include -1 because the square root allows x to be equal to -1 (since sqrt(0) is defined). Then, we go all the way to the right, which is positive infinity, but we have to exclude zero because division by zero isn't allowed. So, the domain is all real numbers from -1 onwards, except for zero.Let me double-check if x can be negative. If x is between -1 and 0, say x = -0.5, then x + 1 would be 0.5, which is positive, so the square root is defined. And since x isn't zero, the denominator is fine. So, x can indeed be negative as long as it's greater than or equal to -1.What about x = -1? Plugging that in, we get sqrt(-1 + 1) = sqrt(0) = 0, and the denominator is -1, so y would be 0 divided by -1, which is 0. That's a valid value, so x = -1 is included.And for x greater than 0, everything works fine too. The square root is defined, and the denominator isn't zero. So, the only restriction is that x can't be zero.Therefore, combining these thoughts, the domain of x is all real numbers greater than or equal to -1, excluding zero. So, in interval notation, that would be [-1, 0) union (0, ∞).Looking back at the options given:A: x ≥ -1 and x ≠ 0B: x > -1 and x ≠ 0C: x > 1D: x ≥ 1Option A matches my conclusion exactly. Option B says x > -1, which would exclude x = -1, but we established that x = -1 is allowed because the square root of zero is defined. So, B is incorrect. Options C and D are way too restrictive because they only consider x greater than 1 or x greater than or equal to 1, which ignores the valid values of x between -1 and 1, excluding zero.So, I'm confident that the correct answer is A.