💡Okay, so I have this probability problem here: Three fair coins are tossed once. For each head that results, two fair dice are rolled. I need to find the probability that the total sum of all dice rolls is odd. If no dice are rolled, the sum is 0, which is even. The answer choices are A. 1/8, B. 3/16, C. 7/16, and D. 1/2.Hmm, let me try to figure this out step by step. First, I think I should consider all the possible outcomes of tossing three coins. Each coin can land on heads (H) or tails (T), so there are 2^3 = 8 possible outcomes. These are:1. HHH2. HHT3. HTH4. THH5. HTT6. THT7. TTH8. TTTNow, for each of these outcomes, I need to determine how many heads there are because for each head, two dice are rolled. Then, I need to figure out the probability that the sum of all the dice rolled is odd.Let me break it down by the number of heads:1. **0 Heads (TTT):** If all three coins are tails, then no dice are rolled. The sum is 0, which is even. So, in this case, the probability of an odd sum is 0.2. **1 Head (HTT, THT, TTH):** There are three outcomes where exactly one head appears. For each of these, two dice are rolled. I need to find the probability that the sum of two dice is odd. I remember that when rolling two dice, the sum is odd if one die is even and the other is odd. Each die has an equal chance of landing on even or odd. So, the probability that the first die is odd and the second is even is (1/2)*(1/2) = 1/4. Similarly, the probability that the first die is even and the second is odd is also 1/4. Adding these together, the probability of an odd sum when rolling two dice is 1/4 + 1/4 = 1/2. So, for each of these three outcomes with one head, the probability of an odd sum is 1/2.3. **2 Heads (HHT, HTH, THH):** There are three outcomes where exactly two heads appear. For each of these, four dice are rolled (two dice per head). I need to find the probability that the sum of four dice is odd. Hmm, this is a bit trickier. I know that the sum of multiple dice can be odd or even. For four dice, the sum is odd if there are an odd number of odd numbers rolled. Since each die is independent, the probability of each die being odd or even is 1/2. Let me think about the possible number of odd dice: - 1 odd and 3 even - 3 odd and 1 even Both of these scenarios result in an odd sum. The probability of getting exactly k odd dice out of n is given by the binomial probability formula: C(n, k)*(1/2)^k*(1/2)^(n-k) = C(n, k)*(1/2)^n. So, for four dice, the probability of getting exactly 1 odd die is C(4,1)*(1/2)^4 = 4/16 = 1/4. Similarly, the probability of getting exactly 3 odd dice is C(4,3)*(1/2)^4 = 4/16 = 1/4. Adding these together, the probability of an odd sum when rolling four dice is 1/4 + 1/4 = 1/2. So, for each of these three outcomes with two heads, the probability of an odd sum is also 1/2.4. **3 Heads (HHH):** There is only one outcome where all three coins are heads. In this case, six dice are rolled. I need to find the probability that the sum of six dice is odd. Similar to the case with four dice, the sum of six dice is odd if there are an odd number of odd dice. The number of odd dice can be 1, 3, or 5. Again, using the binomial probability formula: - Probability of exactly 1 odd die: C(6,1)*(1/2)^6 = 6/64 = 3/32 - Probability of exactly 3 odd dice: C(6,3)*(1/2)^6 = 20/64 = 5/16 - Probability of exactly 5 odd dice: C(6,5)*(1/2)^6 = 6/64 = 3/32 Adding these together: 3/32 + 5/16 + 3/32 = (3 + 10 + 3)/32 = 16/32 = 1/2. So, even when rolling six dice, the probability of an odd sum is 1/2.Now, let me summarize the probabilities:- 0 Heads (TTT): Probability = 1/8, Probability of odd sum = 0- 1 Head: Probability = 3/8, Probability of odd sum = 1/2- 2 Heads: Probability = 3/8, Probability of odd sum = 1/2- 3 Heads: Probability = 1/8, Probability of odd sum = 1/2To find the total probability of an odd sum, I need to multiply the probability of each case by the probability of an odd sum in that case and then add them all together.So, calculating each term:- For 0 Heads: (1/8)*0 = 0- For 1 Head: (3/8)*(1/2) = 3/16- For 2 Heads: (3/8)*(1/2) = 3/16- For 3 Heads: (1/8)*(1/2) = 1/16Adding these up: 0 + 3/16 + 3/16 + 1/16 = (3 + 3 + 1)/16 = 7/16.So, the total probability of the sum being odd is 7/16.Let me double-check my reasoning. For each number of heads, I calculated the probability of that number of heads and then multiplied it by the probability that the sum of the corresponding number of dice is odd. Since in each case where dice are rolled, the probability of an odd sum is 1/2, except when no dice are rolled, which always results in an even sum. Then, I added up the contributions from each case.Yes, that seems correct. So, the answer should be 7/16, which is option C.

💡Okay, so I have this inequality to solve: (x^{2} - (a + 1)x + a geq 0), where (a) is a real number. Hmm, I need to find the values of (x) that satisfy this inequality. Let me think about how to approach this.First, I remember that quadratic inequalities can often be solved by factoring the quadratic expression and then analyzing the intervals determined by the roots. So, maybe I should try to factor (x^{2} - (a + 1)x + a).Let me see... The quadratic is (x^{2} - (a + 1)x + a). To factor this, I need two numbers that multiply to (a) and add up to (-(a + 1)). Wait, that might be tricky because of the variable (a). Maybe I can factor it differently.Let me try to factor by grouping. Let's see:(x^{2} - (a + 1)x + a)I can rewrite the middle term as (-ax - x), so:(x^{2} - ax - x + a)Now, group the first two terms and the last two terms:((x^{2} - ax) + (-x + a))Factor out an (x) from the first group and a (-1) from the second group:(x(x - a) - 1(x - a))Now, I can factor out the common term ((x - a)):((x - a)(x - 1))Oh, nice! So, the quadratic factors into ((x - a)(x - 1)). That makes things easier.So, the inequality becomes:((x - a)(x - 1) geq 0)Now, I need to determine where this product is greater than or equal to zero. To do that, I should find the critical points where each factor is zero because the sign of the product can change at those points.Setting each factor equal to zero:1. (x - a = 0) ⇒ (x = a)2. (x - 1 = 0) ⇒ (x = 1)So, the critical points are (x = a) and (x = 1). These points divide the real number line into intervals. The sign of the product ((x - a)(x - 1)) will be consistent within each interval.But here's the thing: the position of (a) relative to 1 will affect the intervals. So, I need to consider different cases based on the value of (a).**Case 1: (a > 1)**If (a) is greater than 1, then the critical points are ordered as (1 < a). So, the intervals are:1. ( (-infty, 1) )2. ( (1, a) )3. ( (a, infty) )Now, I'll test each interval to determine the sign of the product in that interval.- **Interval ( (-infty, 1) ):** Choose (x = 0). ((0 - a)(0 - 1) = (-a)(-1) = a). Since (a > 1), this is positive.- **Interval ( (1, a) ):** Choose (x = frac{1 + a}{2}) (the midpoint between 1 and (a)). Let's see, (x - a = frac{1 + a}{2} - a = frac{1 - a}{2}), which is negative because (a > 1). (x - 1 = frac{1 + a}{2} - 1 = frac{a - 1}{2}), which is positive because (a > 1). So, the product is negative.- **Interval ( (a, infty) ):** Choose (x = a + 1). ((a + 1 - a)(a + 1 - 1) = (1)(a) = a), which is positive.So, the product is positive in ( (-infty, 1) ) and ( (a, infty) ), and negative in ( (1, a) ).Since the inequality is (geq 0), we include the points where the product is zero, which are (x = 1) and (x = a).Therefore, the solution set is ( (-infty, 1] cup [a, infty) ).**Case 2: (a = 1)**If (a = 1), then both critical points coincide at (x = 1). The quadratic becomes:((x - 1)^2 geq 0)Since a square is always non-negative, this inequality holds for all real numbers (x). So, the solution set is ( mathbb{R} ).**Case 3: (a < 1)**If (a) is less than 1, then the critical points are ordered as (a < 1). So, the intervals are:1. ( (-infty, a) )2. ( (a, 1) )3. ( (1, infty) )Testing each interval:- **Interval ( (-infty, a) ):** Choose (x = a - 1). ((a - 1 - a)(a - 1 - 1) = (-1)(a - 2)). Since (a < 1), (a - 2) is negative, so the product is positive.- **Interval ( (a, 1) ):** Choose (x = frac{a + 1}{2}). (x - a = frac{a + 1}{2} - a = frac{1 - a}{2}), which is positive because (a < 1). (x - 1 = frac{a + 1}{2} - 1 = frac{a - 1}{2}), which is negative because (a < 1). So, the product is negative.- **Interval ( (1, infty) ):** Choose (x = 2). ((2 - a)(2 - 1) = (2 - a)(1) = 2 - a). Since (a < 1), (2 - a) is positive.So, the product is positive in ( (-infty, a) ) and ( (1, infty) ), and negative in ( (a, 1) ).Including the points where the product is zero, the solution set is ( (-infty, a] cup [1, infty) ).**Summary:**- If (a > 1), the solution is ( (-infty, 1] cup [a, infty) ).- If (a = 1), the solution is all real numbers ( mathbb{R} ).- If (a < 1), the solution is ( (-infty, a] cup [1, infty) ).I think that covers all possible cases for (a). I should double-check my work to make sure I didn't make any mistakes in the interval testing or the factoring.Wait, let me verify the factoring again. The original quadratic is (x^{2} - (a + 1)x + a). Factoring it as ((x - a)(x - 1)) gives (x^{2} - (a + 1)x + a), which matches. So, the factoring is correct.Also, in each case, I tested the intervals correctly by choosing test points and checking the sign of the product. It seems consistent.Therefore, I'm confident that the solution sets I found are correct based on the value of (a).

💡First, I need to determine the radius of the central fountain. Given that the diameter of the fountain is 12 feet, the radius is half of that, which is 6 feet.Next, I'll add the width of the flower garden, which is 10 feet, to the radius of the fountain. This gives me 6 feet + 10 feet = 16 feet.Then, I'll add the width of the walking path, which is 6 feet, to the previous total. This results in 16 feet + 6 feet = 22 feet. This is the total radius of the outer boundary of the walking path.Finally, to find the diameter of the outer boundary, I'll multiply the total radius by 2. So, 22 feet * 2 = 44 feet.Therefore, the diameter of the circle that marks the outer boundary of the walking path is 44 feet.

💡Okay, so I need to find the 30th term of this arithmetic sequence: 3, 7, 11, and so on. Hmm, arithmetic sequences have a common difference between consecutive terms, right? Let me check that first. The first term is 3, and the second term is 7. So, the difference between them is 7 minus 3, which is 4. Let me see if this difference stays the same. The third term is 11, so 11 minus 7 is also 4. Okay, so the common difference, which I think is usually denoted as 'd', is 4.Now, I remember there's a formula to find the nth term of an arithmetic sequence. I think it's something like a_n equals a_1 plus (n minus 1) times d. Let me write that down to make sure I have it right: a_n = a_1 + (n - 1)d.In this case, a_1 is 3, d is 4, and n is 30 because we're looking for the 30th term. So, plugging those values into the formula, it should be a_30 = 3 + (30 - 1) * 4.Let me compute that step by step. First, 30 minus 1 is 29. Then, 29 multiplied by 4. Let me do that multiplication: 29 times 4. Well, 20 times 4 is 80, and 9 times 4 is 36, so 80 plus 36 is 116. So, 29 times 4 is 116.Now, adding that to the first term, which is 3: 3 plus 116. That should be 119. So, the 30th term is 119.Wait, let me double-check my calculations to make sure I didn't make a mistake. Starting with a_1 = 3, d = 4, n = 30. The formula is a_n = a_1 + (n - 1)d. Plugging in, a_30 = 3 + (30 - 1)*4. 30 minus 1 is 29, and 29 times 4 is 116. Adding 3 gives 119. Yeah, that seems right.Just to be thorough, maybe I can check a smaller term to see if the formula works. Let's say the 4th term. According to the sequence, it should be 15 because 3, 7, 11, 15. Using the formula: a_4 = 3 + (4 - 1)*4 = 3 + 12 = 15. That matches, so the formula is working correctly.Another check: the 5th term should be 19. Using the formula: a_5 = 3 + (5 - 1)*4 = 3 + 16 = 19. Perfect, that's correct too.So, since the formula works for the earlier terms, it should be reliable for the 30th term as well. Therefore, I'm confident that the 30th term is 119.

💡Alright, so I have this problem here about an ellipse and a chord with a given midpoint. Let me try to figure it out step by step. The ellipse equation is given as (frac{x^2}{36} + frac{y^2}{9} = 1). The midpoint of the chord is ((4, 2)), and I need to find the slope of the line on which this chord lies. The options are A: 2, B: -2, C: 1/3, D: -1/2.First, I remember that for an ellipse, there's a concept called the equation of the chord with a given midpoint. I think it involves the slope of the chord and the coordinates of the midpoint. Maybe it's something related to the point form or the equation of the tangent? Hmm, not sure. Let me think.Wait, I think there's a formula for the equation of the chord of an ellipse given the midpoint. I recall it's similar to the equation of the tangent but adjusted for the chord. The general equation for an ellipse is (frac{x^2}{a^2} + frac{y^2}{b^2} = 1), where (a) and (b) are the semi-major and semi-minor axes. In this case, (a^2 = 36) so (a = 6), and (b^2 = 9) so (b = 3).The midpoint is ((4, 2)). Let me denote this midpoint as ((h, k)), so (h = 4) and (k = 2). I think the equation of the chord with midpoint ((h, k)) is given by (frac{xx_1}{a^2} + frac{yy_1}{b^2} = frac{h^2}{a^2} + frac{k^2}{b^2}), but I'm not entirely sure if that's correct. Maybe I should derive it.Alternatively, I remember that the slope of the chord can be found using the derivative of the ellipse equation at the midpoint, but wait, that's the slope of the tangent at that point, not the chord. Hmm, so that's different. The slope of the chord isn't the same as the slope of the tangent.Let me think of another approach. Suppose the chord intersects the ellipse at two points, say ((x_1, y_1)) and ((x_2, y_2)). The midpoint of these two points is ((4, 2)), so by the midpoint formula, we have:[frac{x_1 + x_2}{2} = 4 quad text{and} quad frac{y_1 + y_2}{2} = 2]Which simplifies to:[x_1 + x_2 = 8 quad text{and} quad y_1 + y_2 = 4]Now, since both points ((x_1, y_1)) and ((x_2, y_2)) lie on the ellipse, they satisfy the ellipse equation:[frac{x_1^2}{36} + frac{y_1^2}{9} = 1][frac{x_2^2}{36} + frac{y_2^2}{9} = 1]If I subtract these two equations, I get:[frac{x_1^2 - x_2^2}{36} + frac{y_1^2 - y_2^2}{9} = 0]This can be factored as:[frac{(x_1 - x_2)(x_1 + x_2)}{36} + frac{(y_1 - y_2)(y_1 + y_2)}{9} = 0]We already know that (x_1 + x_2 = 8) and (y_1 + y_2 = 4), so substituting these in:[frac{(x_1 - x_2)(8)}{36} + frac{(y_1 - y_2)(4)}{9} = 0]Let me denote the slope of the chord as (m). The slope (m) is given by:[m = frac{y_2 - y_1}{x_2 - x_1}]Notice that (y_1 - y_2 = -(y_2 - y_1)) and (x_1 - x_2 = -(x_2 - x_1)), so we can rewrite the equation as:[frac{-(x_2 - x_1)(8)}{36} + frac{-(y_2 - y_1)(4)}{9} = 0]Factor out the negative signs:[-left( frac{(x_2 - x_1)(8)}{36} + frac{(y_2 - y_1)(4)}{9} right) = 0]Which simplifies to:[frac{(x_2 - x_1)(8)}{36} + frac{(y_2 - y_1)(4)}{9} = 0]Divide both terms by 4 to simplify:[frac{(x_2 - x_1)(2)}{9} + frac{(y_2 - y_1)(1)}{9} = 0]Multiply both sides by 9:[2(x_2 - x_1) + (y_2 - y_1) = 0]Now, let's express this in terms of the slope (m). Since (m = frac{y_2 - y_1}{x_2 - x_1}), we can write (y_2 - y_1 = m(x_2 - x_1)). Substitute this into the equation:[2(x_2 - x_1) + m(x_2 - x_1) = 0]Factor out ((x_2 - x_1)):[(x_2 - x_1)(2 + m) = 0]Now, unless (x_2 = x_1), which would mean a vertical line, but in that case, the slope would be undefined, which isn't one of the options. So, we can divide both sides by ((x_2 - x_1)), giving:[2 + m = 0]Therefore, solving for (m):[m = -2]Wait, but that's not one of the options? Let me check the options again. A: 2, B: -2, C: 1/3, D: -1/2. Oh, B is -2. So, that would be the answer. But wait, in my earlier steps, I might have made a mistake. Let me double-check.Starting from the subtraction of the ellipse equations:[frac{x_1^2 - x_2^2}{36} + frac{y_1^2 - y_2^2}{9} = 0]Factoring:[frac{(x_1 - x_2)(x_1 + x_2)}{36} + frac{(y_1 - y_2)(y_1 + y_2)}{9} = 0]Substituting (x_1 + x_2 = 8) and (y_1 + y_2 = 4):[frac{(x_1 - x_2)(8)}{36} + frac{(y_1 - y_2)(4)}{9} = 0]Simplify the fractions:[frac{8(x_1 - x_2)}{36} = frac{2(x_1 - x_2)}{9}][frac{4(y_1 - y_2)}{9} = frac{4(y_1 - y_2)}{9}]So:[frac{2(x_1 - x_2)}{9} + frac{4(y_1 - y_2)}{9} = 0]Multiply both sides by 9:[2(x_1 - x_2) + 4(y_1 - y_2) = 0]Divide both sides by 2:[(x_1 - x_2) + 2(y_1 - y_2) = 0]Express in terms of slope (m = frac{y_2 - y_1}{x_2 - x_1}):[(x_1 - x_2) + 2(y_1 - y_2) = 0][-(x_2 - x_1) + 2(-(y_2 - y_1)) = 0][-(x_2 - x_1) - 2(y_2 - y_1) = 0][-(x_2 - x_1) = 2(y_2 - y_1)][frac{y_2 - y_1}{x_2 - x_1} = -frac{1}{2}][m = -frac{1}{2}]Oh, wait! So I made a mistake earlier when I thought (m = -2). It's actually (-frac{1}{2}). So the slope is (-frac{1}{2}), which is option D.Let me verify this another way. I remember that for an ellipse, the slope of the chord with midpoint ((h, k)) is given by (-frac{b^2 h}{a^2 k}). Let me check if that formula is correct.Given the ellipse (frac{x^2}{a^2} + frac{y^2}{b^2} = 1), the slope of the chord with midpoint ((h, k)) is indeed (-frac{b^2 h}{a^2 k}). Let me plug in the values:[m = -frac{b^2 h}{a^2 k} = -frac{9 times 4}{36 times 2} = -frac{36}{72} = -frac{1}{2}]Yes, that confirms it. So the slope is (-frac{1}{2}), which is option D.I think my initial mistake was in the algebra when I tried to express the slope. By using the formula, it becomes much clearer. So, the correct answer is D: (-frac{1}{2}).**Final Answer**boxed{D}

💡Alright, so I have this problem about an even function ( f(x) ) that's defined on all real numbers and it's decreasing on the interval ( (0, +infty) ). I need to figure out which inequality holds between ( fleft(-frac{3}{4}right) ) and ( fleft(a^{2} - a + 1right) ). The options are A, B, C, and D, which involve greater than, greater than or equal to, less than, and less than or equal to, respectively.First, since ( f(x) ) is an even function, I remember that means ( f(-x) = f(x) ) for all ( x ). So, ( fleft(-frac{3}{4}right) ) is the same as ( fleft(frac{3}{4}right) ). That simplifies things a bit because now I just need to compare ( fleft(frac{3}{4}right) ) with ( fleft(a^{2} - a + 1right) ).Next, the function is decreasing on ( (0, +infty) ). That means as ( x ) increases, ( f(x) ) decreases. So, if I have two points ( x_1 ) and ( x_2 ) in ( (0, +infty) ) where ( x_1 < x_2 ), then ( f(x_1) > f(x_2) ).Now, I need to figure out where ( a^{2} - a + 1 ) falls in relation to ( frac{3}{4} ). Let me compute ( a^{2} - a + 1 ). Hmm, this looks like a quadratic expression. Maybe I can complete the square to understand its minimum value.Completing the square for ( a^{2} - a + 1 ):- The coefficient of ( a ) is -1, so half of that is -1/2.- Squaring that gives ( left(-frac{1}{2}right)^2 = frac{1}{4} ).- So, ( a^{2} - a + 1 = left(a - frac{1}{2}right)^2 + frac{3}{4} ).Ah, so ( a^{2} - a + 1 ) is always greater than or equal to ( frac{3}{4} ) because the square term ( left(a - frac{1}{2}right)^2 ) is always non-negative. The minimum value occurs when ( a = frac{1}{2} ), giving ( frac{3}{4} ).So, ( a^{2} - a + 1 geq frac{3}{4} ). Since ( f(x) ) is decreasing on ( (0, +infty) ), a larger input gives a smaller output. Therefore, ( fleft(a^{2} - a + 1right) leq fleft(frac{3}{4}right) ).But wait, ( fleft(-frac{3}{4}right) = fleft(frac{3}{4}right) ), so combining these, we have ( fleft(-frac{3}{4}right) geq fleft(a^{2} - a + 1right) ).Let me double-check. If ( a^{2} - a + 1 ) is equal to ( frac{3}{4} ), then ( fleft(a^{2} - a + 1right) = fleft(frac{3}{4}right) ). If it's greater than ( frac{3}{4} ), then since the function is decreasing, ( fleft(a^{2} - a + 1right) ) will be less than ( fleft(frac{3}{4}right) ). So, in all cases, ( fleft(-frac{3}{4}right) ) is greater than or equal to ( fleft(a^{2} - a + 1right) ).Therefore, the correct inequality is ( fleft(-frac{3}{4}right) geqslant fleft(a^{2} - a + 1right) ), which is option B.**Final Answer**boxed{B}