💡First, I need to convert the number 320,000 into scientific notation. Scientific notation requires the number to be expressed in the form ( a times 10^{n} ), where ( a ) is a number between 1 and 10, and ( n ) is an integer.To start, I'll identify the significant figures in 320,000. The significant figures are 3 and 2, as the trailing zeros are placeholders indicating magnitude but are not significant.Next, I'll move the decimal point in 320,000 so that there is only one non-zero digit to the left of the decimal. Moving the decimal point five places to the left transforms 320,000 into 3.2.Now, I'll count the number of places the decimal point has moved. In this case, it has moved 5 places to the left.Finally, I'll write the number in scientific notation by multiplying the adjusted number (3.2) by 10 raised to the power of the number of places the decimal moved (5). This gives me:[ 320,000 = 3.2 times 10^{5} ]Comparing this result with the provided options, the correct answer is option B: ( 3.2 times 10^{5} ).

💡First, calculate the volume of water leaked per second by multiplying the number of drops per second by the volume of each drop: 2 drops/second × 0.05 milliliters/drop = 0.1 milliliters/second.Next, convert the total time from hours to seconds: 4 hours × 3600 seconds/hour = 14400 seconds.Then, determine the total volume of water leaked by multiplying the volume per second by the total time: 0.1 milliliters/second × 14400 seconds = 1440 milliliters.Finally, express the result with two significant figures: 1440 milliliters ≈ 1.4 × 10³ milliliters.

💡First, I need to calculate the cube root of 8. The cube root of 8 is 2 because 2 multiplied by itself three times equals 8.Next, I should evaluate the term 2016 raised to the power of 0. Any non-zero number raised to the power of 0 is 1.Then, I need to find the absolute value of the expression inside the absolute value signs: 1 minus the square root of 4. The square root of 4 is 2, so 1 minus 2 equals -1. The absolute value of -1 is 1.Now, I can substitute these calculated values back into the original expression:- Cube root of 8 is 2- 2016 to the power of 0 is 1- Absolute value of (1 - square root of 4) is 1So the expression becomes:-2 + 1 + 1Finally, I perform the arithmetic operations:-2 + 1 equals -1-1 + 1 equals 0Therefore, the result of the expression is 0.

💡Okay, so I have this inequality to solve: (3x + 4)/(x - 2) ≥ 4. Hmm, I remember that when dealing with inequalities involving fractions, I need to be careful about the denominator because it can't be zero, and also the sign of the denominator affects the direction of the inequality when I multiply both sides by it. Let's see how to approach this step by step.First, I should probably get rid of the fraction to make things simpler. To do that, I can subtract 4 from both sides so that I have everything on one side. That would give me:(3x + 4)/(x - 2) - 4 ≥ 0Now, to combine these terms into a single fraction, I need a common denominator. The denominator is already (x - 2), so I can write 4 as 4*(x - 2)/(x - 2). Let's do that:(3x + 4)/(x - 2) - 4*(x - 2)/(x - 2) ≥ 0Now, combine the numerators:[3x + 4 - 4(x - 2)] / (x - 2) ≥ 0Let me simplify the numerator:3x + 4 - 4x + 8 = (3x - 4x) + (4 + 8) = -x + 12So now the inequality becomes:(-x + 12)/(x - 2) ≥ 0Hmm, it's often easier to work with positive coefficients for x, so maybe I can factor out a negative sign from the numerator:-(x - 12)/(x - 2) ≥ 0If I multiply both sides of the inequality by -1, I have to remember to flip the inequality sign:(x - 12)/(x - 2) ≤ 0Alright, now I have (x - 12)/(x - 2) ≤ 0. To solve this, I need to find the values of x where this fraction is less than or equal to zero. First, let's identify the critical points where the numerator or denominator is zero because these points divide the number line into intervals that we can test.The numerator is zero when x - 12 = 0, so x = 12.The denominator is zero when x - 2 = 0, so x = 2.These critical points divide the number line into three intervals:1. x < 22. 2 < x < 123. x > 12Now, I need to test each interval to see where the inequality holds true.Let's start with the first interval: x < 2. I'll pick a test point, say x = 0.Plugging into (x - 12)/(x - 2):(0 - 12)/(0 - 2) = (-12)/(-2) = 6, which is positive. Since we're looking for where the expression is ≤ 0, this interval doesn't satisfy the inequality.Next, the interval 2 < x < 12. Let's choose x = 5.Plugging into (5 - 12)/(5 - 2) = (-7)/3 ≈ -2.333, which is negative. This satisfies the inequality.Finally, the interval x > 12. Let's pick x = 13.Plugging into (13 - 12)/(13 - 2) = 1/11 ≈ 0.09, which is positive. Again, this doesn't satisfy the inequality.Now, we should also check the critical points themselves:At x = 2, the original expression is undefined because the denominator becomes zero. So, x = 2 is not included in the solution.At x = 12, plugging into (12 - 12)/(12 - 2) = 0/10 = 0, which satisfies the inequality since it's equal to zero. So, x = 12 is included.Putting it all together, the solution is the interval where the expression is negative or zero, which is 2 < x ≤ 12.But wait, let me double-check my steps to make sure I didn't make any mistakes.1. Started with (3x + 4)/(x - 2) ≥ 4.2. Subtracted 4 to get (3x + 4 - 4(x - 2))/(x - 2) ≥ 0.3. Simplified numerator to -x + 12.4. Factored out negative to get -(x - 12)/(x - 2) ≥ 0.5. Multiplied both sides by -1, flipping inequality to (x - 12)/(x - 2) ≤ 0.6. Identified critical points at x = 2 and x = 12.7. Tested intervals and found that 2 < x ≤ 12 satisfies the inequality.Everything seems to check out. So, the solution set is all real numbers between 2 and 12, including 12 but not including 2.**Final Answer**The solution set of the inequality is boxed{(2, 12]}.

💡Okay, so I have this problem about an arithmetic sequence and a function, and I need to find the sum of the function applied to each term of the sequence. Let me try to break it down step by step.First, the problem says that we have an arithmetic sequence {a_n} with a non-zero common difference. The sum of the first n terms is S_n, and specifically, S_8 is given as 4π. So, S_8 = 4π. I remember that the sum of the first n terms of an arithmetic sequence can be calculated using the formula:S_n = (n/2)(2a_1 + (n - 1)d)where a_1 is the first term and d is the common difference. Since n is 8 here, plugging that in:S_8 = (8/2)(2a_1 + 7d) = 4(2a_1 + 7d) = 8a_1 + 28dAnd this is equal to 4π. So,8a_1 + 28d = 4πI can simplify this equation by dividing both sides by 4:2a_1 + 7d = πHmm, okay. So that's one equation involving a_1 and d. But I don't know if I'll need to find a_1 or d specifically, or if there's another way to approach this.Next, the function given is f(x) = cos x (2 sin x + 1). So, f(x) is a product of cosine and a linear combination of sine. I need to compute the sum f(a_1) + f(a_2) + ... + f(a_8).Let me write out f(x) again:f(x) = cos x (2 sin x + 1) = 2 cos x sin x + cos xWait, 2 cos x sin x is equal to sin(2x), right? Because sin(2x) = 2 sin x cos x. So, f(x) can be rewritten as:f(x) = sin(2x) + cos xThat might be helpful because working with sine and cosine functions might have some properties that can simplify the sum.So, f(a_k) = sin(2a_k) + cos(a_k)Therefore, the sum f(a_1) + f(a_2) + ... + f(a_8) is equal to:[sin(2a_1) + sin(2a_2) + ... + sin(2a_8)] + [cos(a_1) + cos(a_2) + ... + cos(a_8)]So, I have two separate sums: one involving sine terms and the other involving cosine terms.Let me consider the arithmetic sequence {a_n}. Since it's arithmetic, each term can be written as:a_k = a_1 + (k - 1)dfor k = 1, 2, ..., 8.So, the terms are:a_1, a_1 + d, a_1 + 2d, ..., a_1 + 7dGiven that, the sum S_8 is 4π, which we already used to get 2a_1 + 7d = π.I wonder if there's a relationship between the terms a_k and a_{9 - k} because sometimes in arithmetic sequences, terms equidistant from the ends have some symmetry.Let's see: a_1 + a_8 = a_1 + (a_1 + 7d) = 2a_1 + 7d = π, as we have from earlier.Similarly, a_2 + a_7 = (a_1 + d) + (a_1 + 6d) = 2a_1 + 7d = πSame with a_3 + a_6 and a_4 + a_5. So, each pair adds up to π.That seems useful. Maybe I can pair the terms when computing the sum.Let me try to compute the sum of f(a_k) from k=1 to 8 by pairing terms.So, pair (a_1, a_8), (a_2, a_7), (a_3, a_6), (a_4, a_5).For each pair, let's compute f(a_k) + f(a_{9 - k}).Let's take the first pair: a_1 and a_8.f(a_1) + f(a_8) = [sin(2a_1) + cos(a_1)] + [sin(2a_8) + cos(a_8)]But since a_8 = π - a_1 (because a_1 + a_8 = π), we can substitute:sin(2a_8) = sin(2(π - a_1)) = sin(2π - 2a_1) = -sin(2a_1)Similarly, cos(a_8) = cos(π - a_1) = -cos(a_1)So, f(a_8) = sin(2a_8) + cos(a_8) = -sin(2a_1) - cos(a_1)Therefore, f(a_1) + f(a_8) = [sin(2a_1) + cos(a_1)] + [-sin(2a_1) - cos(a_1)] = 0Wow, that's zero. Interesting.Let me check another pair to see if this holds. Take a_2 and a_7.Similarly, a_2 + a_7 = π, so a_7 = π - a_2.f(a_2) + f(a_7) = [sin(2a_2) + cos(a_2)] + [sin(2a_7) + cos(a_7)]Again, sin(2a_7) = sin(2(π - a_2)) = sin(2π - 2a_2) = -sin(2a_2)cos(a_7) = cos(π - a_2) = -cos(a_2)So, f(a_7) = -sin(2a_2) - cos(a_2)Thus, f(a_2) + f(a_7) = [sin(2a_2) + cos(a_2)] + [-sin(2a_2) - cos(a_2)] = 0Same result.I think this pattern holds for all pairs. So, each pair (a_k, a_{9 - k}) sums to zero.Since there are four such pairs in the eight terms, the total sum would be 4 * 0 = 0.Therefore, the value of f(a_1) + f(a_2) + ... + f(a_8) is 0.But wait, let me make sure I didn't miss anything. Let me consider the function f(x) again.f(x) = cos x (2 sin x + 1) = 2 sin x cos x + cos x = sin(2x) + cos xSo, when I pair a_k and a_{9 - k}, each pair's f(a_k) + f(a_{9 - k}) is zero because:sin(2a_k) + sin(2a_{9 - k}) = sin(2a_k) + sin(2(π - a_k)) = sin(2a_k) - sin(2a_k) = 0Similarly, cos(a_k) + cos(a_{9 - k}) = cos(a_k) + cos(π - a_k) = cos(a_k) - cos(a_k) = 0So, both components cancel out, leading to zero for each pair.Therefore, adding all four pairs gives zero.Hence, the total sum is zero.I think that makes sense. I don't see any mistakes in the reasoning. Each pair cancels out because of the properties of sine and cosine functions when their arguments add up to π. So, the sum is zero.**Final Answer**boxed{A}

💡Okay, so I have this equation here: (x - 3)(3x + 11) = x² - 14x + 48. I need to find the two solutions, r and s, and then compute (r + 4)(s + 4). Hmm, let me think about how to approach this step by step.First, I remember that to solve an equation like this, it's usually helpful to bring all terms to one side so that I can have a standard quadratic equation. That way, I can use the quadratic formula or factorization to find the roots. So, let me start by expanding the left side of the equation.Expanding (x - 3)(3x + 11): I'll use the distributive property, also known as the FOIL method for binomials. First, multiply x by 3x, which gives 3x². Then, x multiplied by 11 is 11x. Next, -3 multiplied by 3x is -9x, and finally, -3 multiplied by 11 is -33. So, putting it all together, the expansion is:3x² + 11x - 9x - 33.Now, let me combine like terms. The 11x and -9x can be combined:11x - 9x = 2x.So, the left side simplifies to:3x² + 2x - 33.Now, the equation is:3x² + 2x - 33 = x² - 14x + 48.To get all terms on one side, I'll subtract x² - 14x + 48 from both sides:3x² + 2x - 33 - (x² - 14x + 48) = 0.Let me distribute the negative sign to each term inside the parentheses:3x² + 2x - 33 - x² + 14x - 48 = 0.Now, combine like terms:3x² - x² = 2x²,2x + 14x = 16x,-33 - 48 = -81.So, the equation simplifies to:2x² + 16x - 81 = 0.Hmm, this is a quadratic equation in the form ax² + bx + c = 0, where a = 2, b = 16, and c = -81. To find the roots, I can use the quadratic formula:x = [-b ± √(b² - 4ac)] / (2a).Plugging in the values:x = [-16 ± √(16² - 4*2*(-81))] / (2*2).Let me compute the discriminant first:16² = 256,4*2 = 8,8*(-81) = -648,So, b² - 4ac = 256 - (-648) = 256 + 648 = 904.So, the square root of 904 is √904. Let me see if 904 can be simplified. Breaking it down:904 divided by 4 is 226, so √904 = √(4*226) = 2√226.So, the roots are:x = [-16 ± 2√226] / 4.I can factor out a 2 in the numerator:x = [2*(-8 ± √226)] / 4 = (-8 ± √226)/2.Simplifying further:x = -4 ± (√226)/2.Wait, that doesn't seem right. Let me check my steps again.Wait, when I had x = [-16 ± 2√226]/4, I can factor out a 2 from numerator and denominator:x = [2*(-8 ± √226)] / 4 = (-8 ± √226)/2.Yes, that's correct. So, the two solutions are:r = (-8 + √226)/2,s = (-8 - √226)/2.Alternatively, I can write them as:r = -4 + (√226)/2,s = -4 - (√226)/2.But maybe it's better to keep them as (-8 ± √226)/2 for now.Now, the question asks for (r + 4)(s + 4). Let me compute that.First, let's compute r + 4 and s + 4.r + 4 = (-8 + √226)/2 + 4 = (-8 + √226)/2 + 8/2 = (-8 + √226 + 8)/2 = √226/2.Similarly, s + 4 = (-8 - √226)/2 + 4 = (-8 - √226)/2 + 8/2 = (-8 - √226 + 8)/2 = (-√226)/2.So, (r + 4)(s + 4) = (√226/2) * (-√226/2).Multiplying these together:(√226 * -√226) / (2 * 2) = (-226)/4.Wait, that simplifies to -226/4, which is -113/2.But that doesn't seem right because when I computed earlier, I thought it was -226. Let me check my steps again.Wait, maybe I made a mistake in simplifying r + 4 and s + 4.Let me try another approach. Instead of computing r + 4 and s + 4 separately, maybe I can use Vieta's formulas, which relate the sum and product of the roots of a quadratic equation.Given a quadratic equation ax² + bx + c = 0, the sum of the roots r + s = -b/a, and the product rs = c/a.In our equation, 2x² + 16x - 81 = 0, so a = 2, b = 16, c = -81.Thus, r + s = -16/2 = -8,and rs = -81/2.Now, we need to find (r + 4)(s + 4).Let me expand this expression:(r + 4)(s + 4) = rs + 4r + 4s + 16.We can rewrite this as rs + 4(r + s) + 16.We already know rs and r + s from Vieta's formulas:rs = -81/2,r + s = -8.Plugging these into the expression:(r + 4)(s + 4) = (-81/2) + 4*(-8) + 16.Let me compute each term:First term: -81/2,Second term: 4*(-8) = -32,Third term: 16.So, adding them together:-81/2 - 32 + 16.Let me convert all terms to have the same denominator to make it easier. The denominator is 2.-81/2 - 32 = -81/2 - 64/2 = (-81 - 64)/2 = -145/2.Then, adding 16, which is 32/2:-145/2 + 32/2 = (-145 + 32)/2 = -113/2.So, (r + 4)(s + 4) = -113/2.Wait, earlier I thought it was -226, but using Vieta's formula, I got -113/2. Which one is correct?Let me check my first approach again. I had:r = (-8 + √226)/2,s = (-8 - √226)/2.Then, r + 4 = (-8 + √226)/2 + 4 = (-8 + √226 + 8)/2 = √226/2.Similarly, s + 4 = (-8 - √226)/2 + 4 = (-8 - √226 + 8)/2 = (-√226)/2.Multiplying them: (√226/2) * (-√226/2) = (-226)/4 = -113/2.Yes, so both methods give the same result: -113/2.But wait, in my initial thought process, I thought it was -226. That must have been a mistake.So, the correct answer is -113/2.But let me double-check the quadratic equation. Maybe I made a mistake in expanding or simplifying.Original equation: (x - 3)(3x + 11) = x² - 14x + 48.Expanding left side: 3x² + 11x - 9x - 33 = 3x² + 2x - 33.Right side: x² - 14x + 48.Bringing all terms to left: 3x² + 2x - 33 - x² + 14x - 48 = 0.Simplify: 2x² + 16x - 81 = 0.Yes, that's correct.So, quadratic equation is 2x² + 16x - 81 = 0.Thus, a = 2, b = 16, c = -81.Vieta's formulas: r + s = -16/2 = -8, rs = -81/2.Then, (r + 4)(s + 4) = rs + 4(r + s) + 16 = (-81/2) + 4*(-8) + 16.Compute:-81/2 -32 +16 = -81/2 -16 = (-81 -32)/2 = -113/2.Yes, that's correct.Alternatively, using the roots:r = (-8 + √226)/2,s = (-8 - √226)/2.Then, r + 4 = (-8 + √226)/2 + 8/2 = (√226)/2,s + 4 = (-8 - √226)/2 + 8/2 = (-√226)/2.Multiplying: (√226)/2 * (-√226)/2 = (-226)/4 = -113/2.So, both methods confirm that (r + 4)(s + 4) = -113/2.Wait, but in my initial thought process, I thought it was -226. That must have been a miscalculation. I must have forgotten to divide by 4 when multiplying the two terms.So, the correct answer is -113/2.But let me check once more to be sure.Alternatively, I can compute (r + 4)(s + 4) directly using the quadratic equation.Let me denote y = x + 4. Then, x = y - 4.Substitute x = y - 4 into the original equation:(x - 3)(3x + 11) = x² - 14x + 48.So, (y - 4 - 3)(3(y - 4) + 11) = (y - 4)² - 14(y - 4) + 48.Simplify left side:(y - 7)(3y - 12 + 11) = (y - 7)(3y - 1).Right side:(y² - 8y + 16) - 14y + 56 + 48.Simplify right side:y² - 8y + 16 -14y +56 +48 = y² -22y + 120.Left side: (y - 7)(3y - 1) = 3y² - y -21y +7 = 3y² -22y +7.So, equation becomes:3y² -22y +7 = y² -22y +120.Subtract y² -22y +120 from both sides:2y² +0y -113 = 0.So, 2y² -113 = 0.Thus, y² = 113/2.So, y = ±√(113/2).But y = x + 4, so the product (r +4)(s +4) is y1 * y2.From the equation 2y² -113 = 0, the product of the roots y1 * y2 = c/a = -113/2.Wait, but in this case, the equation is 2y² -113 = 0, which can be written as 2y² + 0y -113 = 0.So, using Vieta's formula, y1 * y2 = c/a = -113/2.But y1 and y2 are (r +4) and (s +4), so (r +4)(s +4) = -113/2.Yes, that's consistent with the previous results.Therefore, the correct answer is -113/2.I think I initially made a mistake by not properly simplifying the terms when I did the first method, but upon checking with Vieta's formula and substitution, it's clear that the correct answer is -113/2.