💡First, I need to determine the volume of oil that was pumped out of the stationary tank when the oil level dropped by 0.025 feet. This can be calculated by finding the cross-sectional area of the stationary tank and multiplying it by the change in height.The stationary tank has a radius of 100 feet. The cross-sectional area (A) of a cylinder is given by the formula:[ A = pi r^2 ]Substituting the radius:[ A = pi times 100^2 = 10000pi text{ square feet} ]The volume of oil pumped out (V_pumped) is:[ V_{text{pumped}} = A times Delta h = 10000pi times 0.025 = 250pi text{ cubic feet} ]Next, I need to find the radius of the oil truck's tank that has the same volume (250π cubic feet) and a height of 10 feet. The volume of a cylinder is given by:[ V = pi r^2 h ]Setting the volumes equal:[ 250pi = pi r_{text{truck}}^2 times 10 ]Dividing both sides by π:[ 250 = r_{text{truck}}^2 times 10 ]Dividing both sides by 10:[ 25 = r_{text{truck}}^2 ]Taking the square root of both sides:[ r_{text{truck}} = sqrt{25} = 5 text{ feet} ]Therefore, the radius of the oil truck's tank is 5 feet.

💡To determine the number of possible subsets ( A = {a_1, a_2, a_3} ) of ( S = {1, 2, 3, 4, 5, 6, 7, 8, 9} ) such that ( a_1 < a_2 < a_3 ) and ( a_3 - a_2 leq 6 ), we can follow these steps:1. **Calculate the total number of 3-element subsets of ( S ):** The total number of ways to choose 3 elements from 9 is given by the combination formula: [ binom{9}{3} = frac{9!}{3!(9-3)!} = 84 ]2. **Identify subsets that violate the condition ( a_3 - a_2 leq 6 ):** We need to find subsets where ( a_3 - a_2 > 6 ). The only such subset is ( {1, 2, 9} ) because ( 9 - 2 = 7 ), which is greater than 6.3. **Subtract the invalid subset from the total:** Since only one subset violates the condition, we subtract it from the total number of subsets: [ 84 - 1 = 83 ]Therefore, the number of valid subsets ( A ) is (boxed{83}).

💡Okay, so I have this problem where I need to measure exactly 8 liters of water using two buckets. One bucket can hold 10 liters, and the other can hold 6 liters. I'm near a river, so I can fill the buckets as much as I want, but I need to end up with exactly 8 liters in one of the buckets. Hmm, let's see how I can approach this.First, I think I should start by filling one of the buckets completely. Maybe I'll fill the 10-liter bucket first since it's larger. So, I take the 10-liter bucket and fill it up to the brim. Now, I have 10 liters in the big bucket and 0 liters in the smaller one.Next, I need to get some water into the smaller bucket. So, I'll pour water from the 10-liter bucket into the 6-liter bucket until the smaller one is full. After doing that, the 10-liter bucket will have 10 - 6 = 4 liters left, and the 6-liter bucket will be full with 6 liters.Now, I have 4 liters in the 10-liter bucket and 6 liters in the 6-liter bucket. But I need to get to 8 liters, so maybe I should empty the smaller bucket back into the river to reset it. That way, I can use it again to measure more water.After emptying the 6-liter bucket, I have 4 liters in the 10-liter bucket and 0 liters in the 6-liter bucket. Now, I can pour the remaining 4 liters from the 10-liter bucket into the 6-liter bucket. So, the 10-liter bucket is now empty, and the 6-liter bucket has 4 liters.Now, I need to get more water into the 10-liter bucket. I'll fill it up again from the river, so it's full with 10 liters, and the 6-liter bucket still has 4 liters.Finally, I'll pour water from the 10-liter bucket into the 6-liter bucket until the smaller one is full. Since the 6-liter bucket already has 4 liters, it can only take 2 more liters to be full. So, I pour 2 liters from the 10-liter bucket into the 6-liter bucket. This leaves me with 10 - 2 = 8 liters in the 10-liter bucket, which is exactly what I needed.Wait, let me double-check that. I started with 10 liters, poured 6 liters into the smaller bucket, leaving me with 4 liters. Then I emptied the smaller bucket, poured the remaining 4 liters into it, filled the big bucket again, and poured 2 liters into the smaller one to make it full. Yes, that leaves me with 8 liters in the big bucket. That makes sense.I think that works! So, by strategically filling and transferring water between the two buckets, I can measure out exactly 8 liters.

💡Okay, so I have this polynomial ( f(x) = 12 + 35x - 8x^2 + 6x^4 + 5x^5 + 3x^6 ) and I need to evaluate it at ( x = -4 ) using Horner's method. Hmm, I remember Horner's method is a way to evaluate polynomials more efficiently by breaking them down into nested multiplications and additions. But I'm a bit fuzzy on the exact steps, so I need to think carefully.First, let me write down the polynomial again to make sure I have it right: ( f(x) = 12 + 35x - 8x^2 + 6x^4 + 5x^5 + 3x^6 ). Wait, I notice that the polynomial skips some degrees. There's no ( x^3 ) term, right? So, when arranging the terms in descending order of powers, it should be ( 3x^6 + 5x^5 + 6x^4 + 0x^3 - 8x^2 + 35x + 12 ). That makes sense because Horner's method requires the polynomial to be expressed in a nested form, which means all degrees should be accounted for, even if their coefficients are zero.So, let me rewrite the polynomial with all degrees explicitly stated:( f(x) = 3x^6 + 5x^5 + 6x^4 + 0x^3 - 8x^2 + 35x + 12 ).Now, to apply Horner's method, I need to express this polynomial in a nested form. The general form for Horner's method is:( f(x) = (...((a_n x + a_{n-1})x + a_{n-2})x + ...)x + a_0 ).So, starting from the highest degree term, which is ( 3x^6 ), I can build the nested expression step by step.Let me list the coefficients in order from the highest degree to the constant term:- ( a_6 = 3 ) (coefficient of ( x^6 ))- ( a_5 = 5 ) (coefficient of ( x^5 ))- ( a_4 = 6 ) (coefficient of ( x^4 ))- ( a_3 = 0 ) (coefficient of ( x^3 ))- ( a_2 = -8 ) (coefficient of ( x^2 ))- ( a_1 = 35 ) (coefficient of ( x ))- ( a_0 = 12 ) (constant term)Now, I can set up the nested multiplication:Start with the highest coefficient, which is 3.1. Multiply by x and add the next coefficient: ( 3x + 5 ).2. Multiply the result by x and add the next coefficient: ( (3x + 5)x + 6 ).3. Multiply the result by x and add the next coefficient: ( ((3x + 5)x + 6)x + 0 ).4. Multiply the result by x and add the next coefficient: ( (((3x + 5)x + 6)x + 0)x - 8 ).5. Multiply the result by x and add the next coefficient: ( ((((3x + 5)x + 6)x + 0)x - 8)x + 35 ).6. Multiply the result by x and add the constant term: ( (((((3x + 5)x + 6)x + 0)x - 8)x + 35)x + 12 ).So, the nested form is:( f(x) = (((((3x + 5)x + 6)x + 0)x - 8)x + 35)x + 12 ).Now, I need to evaluate this at ( x = -4 ). Let me denote each step as ( v_0, v_1, v_2, ldots, v_6 ), where each ( v_i ) corresponds to the value after each multiplication and addition step.Let me write down each step:1. ( v_0 = 3 ) (the leading coefficient)2. ( v_1 = v_0 times (-4) + 5 )3. ( v_2 = v_1 times (-4) + 6 )4. ( v_3 = v_2 times (-4) + 0 )5. ( v_4 = v_3 times (-4) - 8 )6. ( v_5 = v_4 times (-4) + 35 )7. ( v_6 = v_5 times (-4) + 12 )The question asks for ( V_3 ), which is ( v_3 ) in my notation. So, I need to compute up to ( v_3 ).Let me compute each step one by one:1. ( v_0 = 3 )2. ( v_1 = 3 times (-4) + 5 = -12 + 5 = -7 )3. ( v_2 = -7 times (-4) + 6 = 28 + 6 = 34 )4. ( v_3 = 34 times (-4) + 0 = -136 + 0 = -136 )Wait, but looking back at the polynomial, I have a 0 coefficient for ( x^3 ), so when I add 0, it doesn't change the value. So, ( v_3 = -136 ).But hold on, the options given are A: -144, B: -136, C: -57, D: 34. So, ( v_3 = -136 ) is option B. Hmm, but in the initial problem statement, the polynomial is written as ( f(x) = 12 + 35x - 8x^2 + 6x^4 + 5x^5 + 3x^6 ). Wait, is that correct? Because when I rewrote it, I included all degrees, but maybe the original polynomial doesn't have the ( x^3 ) term, which is why it's zero.But in my calculation, ( v_3 = -136 ), which is option B. However, in the initial analysis provided earlier, the assistant got ( v_3 = -57 ). That seems conflicting. Let me check my steps again.Wait, perhaps I made a mistake in the number of steps. Let me recount the coefficients:The polynomial is ( 3x^6 + 5x^5 + 6x^4 + 0x^3 - 8x^2 + 35x + 12 ). So, the coefficients are 3, 5, 6, 0, -8, 35, 12. That's 7 coefficients, so the number of steps in Horner's method should be 6, right? Because starting from ( v_0 ), each step corresponds to a coefficient.Wait, let me see:- ( v_0 = 3 )- ( v_1 = 3*(-4) + 5 = -12 + 5 = -7 ) (after first coefficient)- ( v_2 = -7*(-4) + 6 = 28 + 6 = 34 ) (after second coefficient)- ( v_3 = 34*(-4) + 0 = -136 + 0 = -136 ) (after third coefficient)- ( v_4 = -136*(-4) -8 = 544 -8 = 536 ) (after fourth coefficient)- ( v_5 = 536*(-4) +35 = -2144 +35 = -2109 ) (after fifth coefficient)- ( v_6 = -2109*(-4) +12 = 8436 +12 = 8448 ) (final value)Wait, but the question is asking for ( V_3 ), which is after the third coefficient. So, according to my calculation, ( V_3 = -136 ), which is option B. But in the initial analysis, the assistant got ( V_3 = -57 ). That's a discrepancy. Maybe I miscounted the steps.Wait, let me check the initial analysis again. The assistant wrote:"Given: ( f(x)=12+35x-8x^2+6x^4+5x^5+3x^6=(((((3x+5)x+6)x+0)x-8)x+35)x+12 ), When ( x=-4 ), Therefore, ( v_0=3 ), ( v_1=3×(-4)+5=-7 ), ( v_2=-7×(-4)+6=34 ), ( v_3=34×(-4)+0=-57 )."Wait, in the assistant's calculation, ( v_3 = 34*(-4) + 0 = -136 ), but they wrote -57. That seems like a miscalculation. Because 34*(-4) is -136, not -57. So, the assistant made a mistake there.But in my calculation, ( v_3 = -136 ), which is option B. However, the options given are A: -144, B: -136, C: -57, D: 34. So, according to my calculation, the correct answer should be B: -136.But wait, let me double-check my steps because sometimes it's easy to miscount the coefficients or the steps.Starting with ( v_0 = 3 ).1. Multiply by x (-4) and add 5: ( 3*(-4) +5 = -12 +5 = -7 ) (v1)2. Multiply by x (-4) and add 6: ( -7*(-4) +6 = 28 +6 =34 ) (v2)3. Multiply by x (-4) and add 0: ( 34*(-4) +0 = -136 +0 = -136 ) (v3)Yes, that seems correct. So, ( v_3 = -136 ), which is option B.But in the initial analysis, the assistant got -57, which is incorrect because 34*(-4) is -136, not -57. So, I think the correct answer is B: -136.Wait, but let me check if the polynomial was written correctly. The original polynomial is ( f(x) = 12 + 35x -8x^2 +6x^4 +5x^5 +3x^6 ). So, it's missing the ( x^3 ) term, which is why we have a 0 coefficient there. So, in the nested form, after the 6x^4 term, we have to include the 0x^3 term, which is why in the nested form, it's (((((3x +5)x +6)x +0)x -8)x +35)x +12.So, in the calculation, after v2=34, we multiply by x (-4) and add 0, which gives v3=-136. So, yes, that's correct.Therefore, the correct answer should be B: -136.But wait, the initial assistant's answer was C: -57, which is incorrect. So, maybe the question is asking for a different ( V_3 ). Let me check the problem statement again.The problem says: "the value of ( V_3 ) is (    )". So, it's referring to the third value in the Horner's computation, which is after processing the third coefficient. So, in my calculation, that's v3=-136.But perhaps the question is using a different indexing. Maybe they start counting from v1 instead of v0. Let me see.If v0 is the first step, then v3 would be the fourth step. Wait, no, in my calculation, v0 is the initial coefficient, then each step increments the index. So, v0 is before any multiplication, v1 after first multiplication, v2 after second, etc.Alternatively, maybe the question is considering the coefficients starting from the constant term. Let me see.Wait, no, Horner's method is usually applied starting from the highest degree. So, the initial coefficient is the highest degree, which is 3 for x^6.Alternatively, maybe the question is considering the coefficients in reverse order. Let me try that.If I reverse the coefficients, starting from the constant term:12, 35, -8, 0, 6, 5, 3.Then, applying Horner's method in reverse:v0 =12v1=12*(-4) +35= -48 +35=-13v2=-13*(-4) + (-8)=52 -8=44v3=44*(-4) +0= -176 +0=-176v4=-176*(-4) +6=704 +6=710v5=710*(-4) +5= -2840 +5=-2835v6=-2835*(-4) +3=11340 +3=11343But this gives a different result, and the question is asking for V3, which would be -176 in this case, which is not among the options. So, that's probably not the case.Therefore, I think the correct approach is to start from the highest degree, as I did earlier, leading to v3=-136, which is option B.Wait, but in the initial problem statement, the polynomial is written as ( f(x)=12+35x-8x^2+6x^4+5x^5+3x^6 ). So, the degrees are 6,5,4,2,1,0. So, when arranging in order, it's 3x^6 +5x^5 +6x^4 +0x^3 -8x^2 +35x +12. So, the coefficients are 3,5,6,0,-8,35,12.Therefore, in Horner's method, starting from 3, then 5, then 6, then 0, then -8, then 35, then 12.So, the steps are:v0=3v1=3*(-4)+5=-12+5=-7v2=-7*(-4)+6=28+6=34v3=34*(-4)+0=-136+0=-136v4=-136*(-4)+(-8)=544-8=536v5=536*(-4)+35=-2144+35=-2109v6=-2109*(-4)+12=8436+12=8448So, f(-4)=8448.But the question is asking for V3, which is -136, option B.However, in the initial analysis, the assistant got -57, which is incorrect. So, I think the correct answer is B: -136.But wait, let me check if I misread the polynomial. The original polynomial is ( f(x)=12+35x-8x^2+6x^4+5x^5+3x^6 ). So, it's 3x^6 +5x^5 +6x^4 -8x^2 +35x +12. So, the coefficient for x^3 is indeed 0.Therefore, in the nested form, it's (((((3x +5)x +6)x +0)x -8)x +35)x +12.So, when evaluating at x=-4:v0=3v1=3*(-4)+5=-12+5=-7v2=-7*(-4)+6=28+6=34v3=34*(-4)+0=-136+0=-136Yes, that's correct. So, V3=-136, which is option B.Therefore, the correct answer is B: -136.

💡First, I need to determine the original price of the shirt and the price Megan paid after the discount.The original price of the shirt is 22, and Megan paid 16 for it after the sale.To find the amount of the discount, I subtract the sale price from the original price.So, the discount is 22 minus 16, which equals 6.Therefore, the amount of the discount is 6.

💡Okay, so I have this system of equations to solve. Let me write them down clearly:First equation: 3x + 4y = 26Second equation: sqrt(x² + y² - 4x + 2y + 5) + sqrt(x² + y² - 20x - 10y + 125) = 10Hmm, the first equation looks linear, so maybe I can solve for one variable in terms of the other and substitute into the second equation. That seems like a standard approach for systems of equations.Let me try solving the first equation for y. So, 3x + 4y = 26. Subtract 3x from both sides: 4y = 26 - 3x. Then divide both sides by 4: y = (26 - 3x)/4.Alright, so y is expressed in terms of x. Now, I can substitute this expression into the second equation. Let me write that out.So, the second equation becomes:sqrt(x² + [(26 - 3x)/4]^2 - 4x + 2*[(26 - 3x)/4] + 5) + sqrt(x² + [(26 - 3x)/4]^2 - 20x - 10*[(26 - 3x)/4] + 125) = 10Wow, that looks complicated. Maybe I can simplify the expressions inside the square roots first.Let me denote the first square root as A and the second as B for clarity.A = sqrt(x² + [(26 - 3x)/4]^2 - 4x + 2*[(26 - 3x)/4] + 5)B = sqrt(x² + [(26 - 3x)/4]^2 - 20x - 10*[(26 - 3x)/4] + 125)So, A + B = 10.Hmm, maybe instead of expanding all of this, I can interpret the second equation geometrically. The expressions inside the square roots look like distance formulas.Let me rewrite the expressions inside the square roots to see if they represent distances.For A:x² + y² - 4x + 2y + 5Let me complete the squares for x and y.x² - 4x = (x - 2)^2 - 4y² + 2y = (y + 1)^2 - 1So, x² + y² - 4x + 2y + 5 = (x - 2)^2 - 4 + (y + 1)^2 - 1 + 5 = (x - 2)^2 + (y + 1)^2Similarly, for B:x² + y² - 20x - 10y + 125Complete the squares again.x² - 20x = (x - 10)^2 - 100y² - 10y = (y - 5)^2 - 25So, x² + y² - 20x - 10y + 125 = (x - 10)^2 - 100 + (y - 5)^2 - 25 + 125 = (x - 10)^2 + (y - 5)^2Oh, interesting! So, the second equation simplifies to:sqrt[(x - 2)^2 + (y + 1)^2] + sqrt[(x - 10)^2 + (y - 5)^2] = 10So, this is the sum of the distances from the point (x, y) to the points (2, -1) and (10, 5) equals 10.Wait a minute, in geometry, the set of points where the sum of the distances to two fixed points is constant is an ellipse, with the two fixed points as foci. But the constant here is 10. So, is this an ellipse with foci at (2, -1) and (10, 5), and the major axis length 10?But before jumping to conclusions, let me check the distance between the two foci. If the distance between the foci is greater than 10, then there would be no such points. If it's equal to 10, then the ellipse collapses to the line segment between the two foci. If it's less than 10, then it's a proper ellipse.Let me calculate the distance between (2, -1) and (10, 5).Distance = sqrt[(10 - 2)^2 + (5 - (-1))^2] = sqrt[8^2 + 6^2] = sqrt[64 + 36] = sqrt[100] = 10.Oh, so the distance between the two foci is exactly 10, which is equal to the sum of the distances. That means the ellipse collapses to the line segment connecting the two foci. So, the only points (x, y) that satisfy the second equation lie on the line segment between (2, -1) and (10, 5).Therefore, the solution to the system must be the intersection of the line 3x + 4y = 26 with the line segment between (2, -1) and (10, 5).So, first, let me find the equation of the line passing through (2, -1) and (10, 5).The slope m is (5 - (-1))/(10 - 2) = 6/8 = 3/4.So, the equation of the line is y - (-1) = (3/4)(x - 2), which simplifies to y + 1 = (3/4)(x - 2). Therefore, y = (3/4)x - (3/4)*2 - 1 = (3/4)x - 3/2 - 1 = (3/4)x - 5/2.So, the line is y = (3/4)x - 5/2.Now, we have two equations:1. 3x + 4y = 262. y = (3/4)x - 5/2Let me substitute equation 2 into equation 1.3x + 4[(3/4)x - 5/2] = 26Simplify:3x + 3x - 10 = 26Combine like terms:6x - 10 = 26Add 10 to both sides:6x = 36Divide both sides by 6:x = 6Now, substitute x = 6 into equation 2:y = (3/4)*6 - 5/2 = (18/4) - 5/2 = (9/2) - (5/2) = 4/2 = 2So, the solution is x = 6, y = 2.Wait, let me verify this solution in both original equations to make sure.First equation: 3*6 + 4*2 = 18 + 8 = 26. That checks out.Second equation: sqrt[(6)^2 + (2)^2 - 4*6 + 2*2 + 5] + sqrt[(6)^2 + (2)^2 - 20*6 - 10*2 + 125]Calculate each square root separately.First sqrt: 36 + 4 - 24 + 4 + 5 = 36 + 4 = 40; 40 - 24 = 16; 16 + 4 = 20; 20 + 5 = 25. So sqrt(25) = 5.Second sqrt: 36 + 4 - 120 - 20 + 125. 36 + 4 = 40; 40 - 120 = -80; -80 - 20 = -100; -100 + 125 = 25. So sqrt(25) = 5.So, 5 + 5 = 10, which matches the second equation.Therefore, the solution (6, 2) satisfies both equations.I think that's the only solution because the line 3x + 4y = 26 intersects the line segment between (2, -1) and (10, 5) at exactly one point, which is (6, 2). Since the ellipse has collapsed to the line segment, there are no other points to consider.So, I'm confident that (6, 2) is the solution.