💡Alright, so I've got this probability problem here about a radio quiz. Let me read it again to make sure I understand it properly."A radio program has expanded its quiz to consist of 5 multiple-choice questions, each with 3 choices. A contestant wins if he or she gets 3 or more of the questions right. The contestant answers randomly to each question. What is the probability of winning?"Okay, so there are 5 questions, each with 3 possible answers. The contestant is guessing randomly, so the probability of getting any question right is 1/3, and the probability of getting it wrong is 2/3. The contestant needs to get at least 3 questions right to win. I need to find the probability of that happening.Hmm, this sounds like a binomial probability problem. In binomial problems, we have a fixed number of independent trials, each with two possible outcomes: success or failure. Here, each question is a trial, success is getting it right, and failure is getting it wrong. The number of trials is 5, and we want the probability of getting 3, 4, or 5 successes.Right, so the formula for binomial probability is:P(k) = C(n, k) * p^k * (1-p)^(n-k)Where:- P(k) is the probability of getting exactly k successes,- C(n, k) is the combination of n things taken k at a time,- p is the probability of success on a single trial,- n is the number of trials.In this case, n = 5, p = 1/3, and we need to calculate P(3) + P(4) + P(5).Let me compute each of these probabilities step by step.First, let's compute P(3):C(5, 3) is the number of ways to choose 3 successes out of 5 trials. The combination formula is C(n, k) = n! / (k!(n - k)!).So, C(5, 3) = 5! / (3! * (5 - 3)!) = (5 * 4 * 3 * 2 * 1) / [(3 * 2 * 1) * (2 * 1)] = 120 / (6 * 2) = 120 / 12 = 10.Then, p^3 = (1/3)^3 = 1/27.And (1 - p)^(5 - 3) = (2/3)^2 = 4/9.So, P(3) = 10 * (1/27) * (4/9) = 10 * (4/243) = 40/243.Okay, that seems right.Next, P(4):C(5, 4) = 5! / (4! * (5 - 4)!) = 120 / (24 * 1) = 5.p^4 = (1/3)^4 = 1/81.(1 - p)^(5 - 4) = (2/3)^1 = 2/3.So, P(4) = 5 * (1/81) * (2/3) = 5 * (2/243) = 10/243.Alright, moving on to P(5):C(5, 5) = 1, since there's only one way to choose all 5.p^5 = (1/3)^5 = 1/243.(1 - p)^(5 - 5) = (2/3)^0 = 1.So, P(5) = 1 * (1/243) * 1 = 1/243.Now, to find the total probability of winning, I need to add up P(3), P(4), and P(5):Total P = P(3) + P(4) + P(5) = 40/243 + 10/243 + 1/243.Adding those together: 40 + 10 + 1 = 51, so 51/243.Hmm, can this fraction be simplified? Let's see. 51 and 243 are both divisible by 3.51 ÷ 3 = 17.243 ÷ 3 = 81.So, 51/243 simplifies to 17/81.Looking back at the answer choices, option C is 17/81, so that must be the correct answer.Wait a second, let me double-check my calculations to make sure I didn't make any mistakes.For P(3): C(5,3) is 10, (1/3)^3 is 1/27, and (2/3)^2 is 4/9. Multiplying them together: 10 * 1/27 * 4/9 = 10 * 4 / 243 = 40/243. That looks correct.For P(4): C(5,4) is 5, (1/3)^4 is 1/81, and (2/3)^1 is 2/3. Multiplying them: 5 * 1/81 * 2/3 = 10/243. That also seems right.For P(5): C(5,5) is 1, (1/3)^5 is 1/243, and (2/3)^0 is 1. So, 1 * 1/243 * 1 = 1/243. Correct.Adding them up: 40 + 10 + 1 = 51, over 243. Simplifying 51/243 by dividing numerator and denominator by 3 gives 17/81. Yep, that all checks out.I think I've got it right. The probability of winning is 17/81, which is option C.**Final Answer**boxed{C}

💡First, I need to understand what 30 million kilometers represents in numerical form. 30 million kilometers is equal to 30,000,000 km.Next, I need to convert this number into scientific notation. In scientific notation, a number is expressed as a product of a number between 1 and 10 and a power of 10.So, 30,000,000 km can be written as 3 × 10^7 km.This is because 30,000,000 has 7 zeros after the 3, which corresponds to 10 raised to the power of 7.Finally, I compare this result with the given options to determine the correct answer.

💡Okay, so I have this problem where I need to find the maximum possible area of a triangle with sides (a), (b), and (c) that satisfy the inequalities (0 < a leq 1 leq b leq 2 leq c leq 3). Hmm, that seems a bit tricky, but let me try to break it down step by step.First, I remember that for any triangle, the sum of any two sides must be greater than the third side. This is called the triangle inequality. So, I need to make sure that (a + b > c), (a + c > b), and (b + c > a). Given the constraints on (a), (b), and (c), I need to check if these inequalities hold.Given (0 < a leq 1), (1 leq b leq 2), and (2 leq c leq 3), let's see:1. (a + b > c): The smallest (a) can be is just above 0, and the smallest (b) is 1, so the smallest (a + b) is just above 1. But (c) is at least 2, so (a + b) might not be greater than (c). That could be a problem.2. (a + c > b): The smallest (a) is just above 0, and the smallest (c) is 2, so (a + c) is just above 2. The largest (b) is 2, so (a + c) is just barely greater than (b). That might work.3. (b + c > a): The smallest (b) is 1, and the smallest (c) is 2, so (b + c) is 3. The largest (a) is 1, so (b + c) is definitely greater than (a).So, the main issue is the first inequality (a + b > c). Since (c) can be as large as 3, and (a + b) can be at most (1 + 2 = 3), the maximum (a + b) is equal to the maximum (c). That means, to satisfy (a + b > c), (c) must be less than 3 when (a) and (b) are at their maximums.Wait, but if (a = 1) and (b = 2), then (c) must be less than 3 to satisfy (a + b > c). So, (c) can be up to just below 3. But since (c) is given as up to 3, maybe we can approach 3 but not reach it. Hmm, interesting.Now, moving on to the area. I know that the area of a triangle can be calculated using Heron's formula, which is:[text{Area} = sqrt{s(s - a)(s - b)(s - c)}]where (s) is the semi-perimeter:[s = frac{a + b + c}{2}]But Heron's formula might be a bit complicated here because it involves square roots and multiple variables. Maybe there's a simpler way to maximize the area.I also remember that the area of a triangle can be expressed as:[text{Area} = frac{1}{2}ab sin C]where (C) is the angle between sides (a) and (b). To maximize the area, (sin C) should be as large as possible, which is 1 when (C = 90^circ). So, the maximum area occurs when the triangle is right-angled between sides (a) and (b).So, if I can form a right-angled triangle with sides (a) and (b), then the area would be maximized. Let's see if that's possible within the given constraints.If (a = 1) and (b = 2), then the hypotenuse (c) would be:[c = sqrt{a^2 + b^2} = sqrt{1^2 + 2^2} = sqrt{1 + 4} = sqrt{5} approx 2.236]Okay, (sqrt{5}) is approximately 2.236, which is between 2 and 3, so it satisfies the condition (2 leq c leq 3). Great! So, a right-angled triangle with sides (a = 1), (b = 2), and (c = sqrt{5}) is possible.The area of this triangle would be:[text{Area} = frac{1}{2} times 1 times 2 = 1]So, the area is 1. Is this the maximum possible area?Let me check if there's a way to get a larger area. Maybe by choosing different values for (a), (b), and (c).Suppose I take (a = 1), (b = 2), and (c) slightly less than 3. Would that give a larger area? Let's see.If (c) approaches 3, then the triangle becomes very "flat", almost a straight line, which would make the area approach zero. So, that's not helpful.What if I take (a) less than 1? Let's say (a = 0.5), (b = 2), and (c) as large as possible. Then, the area would be:[text{Area} = frac{1}{2} times 0.5 times 2 times sin C = 0.5 times sin C]To maximize this, (sin C) should be 1, so the maximum area would be 0.5, which is less than 1. So, that's worse.What if I take (b) less than 2? Let's say (b = 1.5), (a = 1), and (c = sqrt{1^2 + 1.5^2} = sqrt{1 + 2.25} = sqrt{3.25} approx 1.802). But (c) needs to be at least 2, so this doesn't work.Alternatively, if I take (a = 1), (b = 2), and (c) slightly more than (sqrt{5}), but still less than 3, would that give a larger area? Let's see.Using Heron's formula, let's calculate the area for (a = 1), (b = 2), and (c = 2.5):First, calculate the semi-perimeter:[s = frac{1 + 2 + 2.5}{2} = frac{5.5}{2} = 2.75]Then, the area is:[sqrt{2.75(2.75 - 1)(2.75 - 2)(2.75 - 2.5)} = sqrt{2.75 times 1.75 times 0.75 times 0.25}]Calculating inside the square root:[2.75 times 1.75 = 4.8125][0.75 times 0.25 = 0.1875][4.8125 times 0.1875 approx 0.90234375]So, the area is approximately:[sqrt{0.90234375} approx 0.95]Which is less than 1. So, the area is smaller when (c) is larger than (sqrt{5}).What if I take (c) smaller than (sqrt{5})? Let's say (c = 2), which is the minimum value for (c). Then, the triangle would have sides (a = 1), (b = 2), (c = 2). Let's calculate the area.Using Heron's formula:[s = frac{1 + 2 + 2}{2} = frac{5}{2} = 2.5][text{Area} = sqrt{2.5(2.5 - 1)(2.5 - 2)(2.5 - 2)} = sqrt{2.5 times 1.5 times 0.5 times 0.5}][2.5 times 1.5 = 3.75][0.5 times 0.5 = 0.25][3.75 times 0.25 = 0.9375][sqrt{0.9375} approx 0.9682]Still less than 1. So, the maximum area seems to occur when (c = sqrt{5}), giving an area of 1.Wait, but is there another configuration where the area could be larger? Maybe if the triangle isn't right-angled but has a different angle that gives a larger area?Let me think. The area is (frac{1}{2}ab sin C). To maximize this, (sin C) needs to be as large as possible, which is 1. So, the maximum area is indeed when (C = 90^circ), making the triangle right-angled.Therefore, the maximum area is achieved when (a = 1), (b = 2), and (c = sqrt{5}), giving an area of 1.But just to be thorough, let me check if there's a way to have a larger area by adjusting (a), (b), and (c) differently.Suppose I take (a) slightly less than 1, say (a = 0.9), (b = 2), and (c = sqrt{0.9^2 + 2^2} = sqrt{0.81 + 4} = sqrt{4.81} approx 2.193). Then, the area would be:[text{Area} = frac{1}{2} times 0.9 times 2 = 0.9]Which is less than 1. So, that's worse.Alternatively, if I take (a = 1), (b = 1.9), and (c = sqrt{1^2 + 1.9^2} = sqrt{1 + 3.61} = sqrt{4.61} approx 2.147). Then, the area is:[text{Area} = frac{1}{2} times 1 times 1.9 = 0.95]Still less than 1.What if I take (a = 1), (b = 2), and (c) just slightly less than 3? Let's say (c = 2.9). Then, the area would be:Using Heron's formula:[s = frac{1 + 2 + 2.9}{2} = frac{5.9}{2} = 2.95][text{Area} = sqrt{2.95(2.95 - 1)(2.95 - 2)(2.95 - 2.9)} = sqrt{2.95 times 1.95 times 0.95 times 0.05}][2.95 times 1.95 approx 5.7525][0.95 times 0.05 = 0.0475][5.7525 times 0.0475 approx 0.273][sqrt{0.273} approx 0.522]That's much less than 1. So, definitely worse.What if I take (a = 1), (b = 2), and (c = sqrt{5}) as before? The area is 1, which seems to be the maximum.Wait, another thought: maybe if I don't take (a) and (b) as the legs of the right triangle, but instead take (b) and (c) as the legs? Let's see.If (b = 2) and (c = 3), then (a) would be:[a = sqrt{b^2 + c^2} = sqrt{4 + 9} = sqrt{13} approx 3.606]But (a) is constrained to be (a leq 1), so that's not possible. Therefore, that configuration isn't allowed.Alternatively, if I take (a) and (c) as the legs, then (b) would be:[b = sqrt{a^2 + c^2}]But (b) is constrained to be (1 leq b leq 2), and (c) is (2 leq c leq 3). Let's see if this is possible.Suppose (a = 1) and (c = 2), then (b = sqrt{1 + 4} = sqrt{5} approx 2.236), which is greater than 2, so it violates the constraint (b leq 2). Therefore, that's not allowed.If I take (a = 1) and (c = 2.5), then (b = sqrt{1 + 6.25} = sqrt{7.25} approx 2.692), which is still greater than 2. So, that's not allowed either.Therefore, taking (b) and (c) as legs of a right triangle isn't possible within the given constraints because it would require (b) to be larger than 2, which is not allowed.Similarly, taking (a) and (c) as legs would require (b) to be larger than 2, which is also not allowed.So, the only feasible right-angled triangle is when (a) and (b) are the legs, giving (c = sqrt{5}), which is within the allowed range for (c).Therefore, the maximum area is indeed 1.Just to double-check, let's consider if there's a way to have a non-right-angled triangle with a larger area. For that, the angle between (a) and (b) would need to be greater than 90 degrees, making (sin C) larger than 1, but that's impossible because the maximum value of (sin C) is 1. So, no, a right angle gives the maximum possible area.Therefore, I'm confident that the maximum possible area is 1.

💡First, I need to calculate the sum of the first 100 positive integers that Jo has computed. Using the formula for the sum of an arithmetic series, which is ( S = frac{n(n+1)}{2} ), where ( n = 100 ), I can determine Jo's total.Next, I need to understand how Kate computes her sum. Kate rounds each integer from 1 to 100 to the nearest multiple of 5 before adding them up. I'll examine how rounding works for numbers in different ranges:- Numbers 1-2 round down to 0.- Numbers 3-7 round up to 5.- Numbers 8-12 round up to 10.- This pattern repeats every 5 numbers.For each set of 5 consecutive numbers, the rounded values are 0, 5, 5, 5, and 10, which sum up to 25. Since there are 20 such groups in 100 numbers, Kate's total sum is ( 20 times 25 = 500 ).Finally, to find the positive difference between Jo's sum and Kate's sum, I subtract Kate's total from Jo's total: ( 5050 - 500 = 4550 ).

💡Okay, so I have this math problem here about an operation called "nabla," which is defined as ( g nabla h = g^2 - h^2 ). They gave an example: ( 2 nabla 1 = 2^2 - 1^2 = 4 - 1 = 3 ). That makes sense. It's like subtracting the squares of the two numbers.Now, the problem says that ( g > 0 ) and ( g nabla 6 = 45 ). I need to find the value of ( g ) from the given options: (A) 39, (B) 6, (C) 81, (D) 3, (E) 9.Alright, let's break this down. The operation ( nabla ) takes two numbers, squares each of them, and then subtracts the second square from the first. So, in this case, ( g nabla 6 ) means ( g^2 - 6^2 ). And they told us that this equals 45.So, writing that out as an equation: ( g^2 - 6^2 = 45 ). Let me compute ( 6^2 ) first. That's 36. So, substituting that in, the equation becomes ( g^2 - 36 = 45 ).Now, I need to solve for ( g ). Let's add 36 to both sides of the equation to isolate ( g^2 ). So, ( g^2 = 45 + 36 ). Calculating that, 45 plus 36 is 81. So, ( g^2 = 81 ).To find ( g ), I take the square root of both sides. The square root of 81 is 9, but since they specified that ( g > 0 ), we only consider the positive root. So, ( g = 9 ).Let me double-check to make sure I didn't make any mistakes. If ( g = 9 ), then ( 9 nabla 6 ) should be ( 9^2 - 6^2 = 81 - 36 = 45 ). Yep, that matches the given equation. So, it looks like 9 is the correct answer.Looking back at the options, (E) is 9, so that must be the right choice.**Final Answer**boxed{E}

💡First, determine the number of days John walked the dog in April, excluding the 4 Sundays. April has 30 days, so the number of days John worked is 30 - 4 = 26 days.Next, calculate John's total earnings from walking the dog. He earned 10 per day, so his total earnings are 26 days * 10/day = 260.John spent 50 on books, so the amount he had left after this expense is 260 - 50 = 210.John had 160 left after giving some money to his sister Kaylee. Therefore, the amount he gave to Kaylee is 210 - 160 = 50.Finally, find the ratio of the amount John spent on books to the amount he gave his sister Kaylee. Both amounts are 50, so the ratio is 50:50, which simplifies to 1:1.