answer：First, I need to understand the two transformations to apply to the complex number (-1 + 7i). The first transformation is a (60^circ) rotation counter-clockwise around the origin, and the second is a dilation centered at the origin with a scale factor of 2.For the rotation, multiplying a complex number by (operatorname{cis} 60^circ) (which is (frac{1}{2} + frac{sqrt{3}i}{2})) will rotate it by (60^circ). Then, the dilation by a scale factor of 2 means multiplying the result by 2. Combining these, the overall transformation is equivalent to multiplying the original complex number by (1 + sqrt{3}i).Next, I'll perform the multiplication:[(-1 + 7i)(1 + sqrt{3}i) = (-1)(1) + (-1)(sqrt{3}i) + 7i(1) + 7i(sqrt{3}i)]Calculating each term:[-1 - sqrt{3}i + 7i + 7sqrt{3}i^2]Since (i^2 = -1), this simplifies to:[-1 - sqrt{3}i + 7i - 7sqrt{3}]Combining like terms:[-1 - 7sqrt{3} + (7 - sqrt{3})i]This gives the final complex number after both transformations.

answer：Okay, so Susan wants to mix two types of peanuts: Virginia peanuts and Spanish peanuts. She has 10 pounds of Virginia peanuts that cost 3.50 per pound. She wants to mix these with Spanish peanuts that cost 3.00 per pound to create a mixture that costs 3.40 per pound. The question is asking how many pounds of Spanish peanuts she should use.Alright, let's break this down. First, I need to figure out the total cost of the Virginia peanuts. Since she has 10 pounds at 3.50 per pound, that would be 10 multiplied by 3.50. Let me calculate that: 10 * 3.50 = 35.00. So, the Virginia peanuts cost her 35.00 in total.Now, she's going to mix these with Spanish peanuts. Let's denote the number of pounds of Spanish peanuts she needs as x. Since Spanish peanuts cost 3.00 per pound, the total cost for x pounds would be 3.00 * x, which is 3.00x.The next thing I need to consider is the total weight of the mixture. She's mixing 10 pounds of Virginia peanuts with x pounds of Spanish peanuts, so the total weight will be 10 + x pounds.Susan wants the mixture to cost 3.40 per pound. That means the total cost of the mixture should be 3.40 multiplied by the total weight, which is 3.40 * (10 + x). Let me write that out: 3.40 * (10 + x).Now, I can set up an equation to represent the total cost of the mixture. The total cost of the Virginia peanuts plus the total cost of the Spanish peanuts should equal the total cost of the mixture. So, that would be:35.00 + 3.00x = 3.40 * (10 + x)Okay, now I need to solve for x. Let's expand the right side of the equation first. 3.40 multiplied by 10 is 34.00, and 3.40 multiplied by x is 3.40x. So, the equation becomes:35.00 + 3.00x = 34.00 + 3.40xNow, I want to get all the terms with x on one side and the constants on the other. Let me subtract 3.00x from both sides to get:35.00 = 34.00 + 0.40xNext, I'll subtract 34.00 from both sides to isolate the term with x:35.00 - 34.00 = 0.40xWhich simplifies to:1.00 = 0.40xNow, to solve for x, I'll divide both sides by 0.40:x = 1.00 / 0.40Calculating that gives:x = 2.5So, Susan needs to use 2.5 pounds of Spanish peanuts to achieve the desired mixture.Wait, let me double-check my calculations to make sure I didn't make a mistake. Starting from the equation:35 + 3x = 34 + 3.40xSubtracting 3x from both sides:35 = 34 + 0.40xSubtracting 34:1 = 0.40xDividing by 0.40:x = 2.5Yes, that seems correct. So, Susan should use 2.5 pounds of Spanish peanuts.

answer：First, I need to convert the number 60,000 into scientific notation. Scientific notation requires expressing the number as a value between 1 and 10 multiplied by a power of 10.Starting with 60,000, I can see that moving the decimal point four places to the left will give me 6. Therefore, 60,000 can be written as 6 multiplied by 10 raised to the power of 4.Next, I compare this result with the given options:- Option A: ( 0.6 times 10^{5} ) is not correct because 0.6 is less than 1.- Option B: ( 6 times 10^{4} ) matches my calculation.- Option C: ( 6 times 10^{5} ) is incorrect because the exponent is too high.- Option D: ( 60 times 10^{3} ) is not correct because 60 is greater than 10.Therefore, the correct representation of 60,000 in scientific notation is ( 6 times 10^{4} ), which corresponds to option B.

answer：First, I need to determine the side length of the square. Given that the area of the square is 3600 square units, I can find the side length by taking the square root of the area.Next, since the radius of the circle is equal to the side length of the square, the radius of the circle will also be 60 units.Then, I calculate the length of the rectangle, which is two-fifths of the radius of the circle. Multiplying two-fifths by 60 gives me the length of the rectangle.Finally, with the length and breadth of the rectangle known, I can find the area of the rectangle by multiplying the length by the breadth.

answer：First, I need to evaluate the expression inside the parentheses: (3 - 8), which equals (-5).Next, I'll square the result from the parentheses: ((-5)^2 = 25).Then, I'll multiply this squared value by 6: (6 times 25 = 150).Finally, I'll add 8 to the result of the multiplication: (8 + 150 = 158).

answer：Okay, so I have this problem about triangle XYZ in the Cartesian plane. The area is 50, and I know the coordinates of two of the vertices: X is at (10, 15) and Y is at (20, 17). The third vertex, Z, has coordinates (r, s). There's also a condition that the median to side XY has a slope of -3. I need to find the highest possible value of r + s.First, let me recall what a median is. A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side. So, the median to side XY would be the line segment from point Z to the midpoint of XY.Alright, so I should start by finding the midpoint of XY. The midpoint formula is ((x1 + x2)/2, (y1 + y2)/2). Plugging in the coordinates of X and Y:Midpoint M = ((10 + 20)/2, (15 + 17)/2) = (30/2, 32/2) = (15, 16).Okay, so the midpoint M is at (15, 16). Now, the median from Z to M has a slope of -3. That means the line connecting Z(r, s) and M(15, 16) has a slope of -3.The slope between two points (r, s) and (15, 16) is given by (s - 16)/(r - 15). So, setting that equal to -3:(s - 16)/(r - 15) = -3.Let me solve this equation for s in terms of r. Multiply both sides by (r - 15):s - 16 = -3(r - 15).Expanding the right side:s - 16 = -3r + 45.Adding 16 to both sides:s = -3r + 45 + 16.So, s = -3r + 61.Alright, so that's an equation relating s and r. Now, I need to use the area condition. The area of triangle XYZ is 50. I can use the shoelace formula to express the area in terms of the coordinates of X, Y, and Z.The shoelace formula for the area of a triangle with vertices (x1, y1), (x2, y2), (x3, y3) is:Area = (1/2)|x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|.Plugging in the coordinates:Area = (1/2)|10(17 - s) + 20(s - 15) + r(15 - 17)|.Simplify each term:First term: 10(17 - s) = 170 - 10s.Second term: 20(s - 15) = 20s - 300.Third term: r(15 - 17) = r(-2) = -2r.So, putting it all together:Area = (1/2)|170 - 10s + 20s - 300 - 2r|.Combine like terms:-10s + 20s = 10s.170 - 300 = -130.So, inside the absolute value: 10s - 2r - 130.Therefore, Area = (1/2)|10s - 2r - 130|.We know the area is 50, so:(1/2)|10s - 2r - 130| = 50.Multiply both sides by 2:|10s - 2r - 130| = 100.This means that 10s - 2r - 130 = 100 or 10s - 2r - 130 = -100.So, two equations:1) 10s - 2r - 130 = 1002) 10s - 2r - 130 = -100Let me simplify both.Starting with equation 1:10s - 2r - 130 = 100Add 130 to both sides:10s - 2r = 230Divide both sides by 2:5s - r = 115.Equation 1: 5s - r = 115.Equation 2:10s - 2r - 130 = -100Add 130 to both sides:10s - 2r = 30Divide both sides by 2:5s - r = 15.Equation 2: 5s - r = 15.So, now I have two equations from the area condition:1) 5s - r = 1152) 5s - r = 15But I also have the equation from the median slope:s = -3r + 61.So, I can substitute s = -3r + 61 into both equations and solve for r.Let's start with equation 1:5s - r = 115Substitute s:5(-3r + 61) - r = 115Multiply out:-15r + 305 - r = 115Combine like terms:-16r + 305 = 115Subtract 305 from both sides:-16r = 115 - 305-16r = -190Divide both sides by -16:r = (-190)/(-16) = 190/16 = 95/8 = 11.875.Hmm, 95 divided by 8 is 11.875. So, r = 11.875.Then, s = -3r + 61 = -3*(95/8) + 61.Calculate:-3*(95/8) = -285/8.Convert 61 to eighths: 61 = 488/8.So, s = (-285/8) + (488/8) = (488 - 285)/8 = 203/8 = 25.375.So, one solution is r = 95/8, s = 203/8.Then, r + s = 95/8 + 203/8 = (95 + 203)/8 = 298/8 = 149/4 = 37.25.Wait, that's 37.25. Hmm, that seems higher than the initial approximate value of 29.26. Maybe I made a mistake earlier.Wait, let me check my calculations.Wait, in the initial assistant's solution, they had s = -3r + 61, substituted into the area equation, and got r ≈15.87, s≈13.39, so r+s≈29.26.But in my calculation, I get r = 95/8 = 11.875, s = 203/8 = 25.375, so r + s = 37.25.Wait, that's a big difference. So, perhaps I made a mistake in the area equation.Wait, let me double-check the shoelace formula.The shoelace formula is (1/2)|x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|.So, plugging in:x1 = 10, y1 = 15x2 = 20, y2 = 17x3 = r, y3 = sSo,Area = (1/2)|10*(17 - s) + 20*(s - 15) + r*(15 - 17)|Compute each term:10*(17 - s) = 170 - 10s20*(s - 15) = 20s - 300r*(15 - 17) = -2rSo, adding them together:170 - 10s + 20s - 300 - 2r = (170 - 300) + (-10s + 20s) - 2r = (-130) + (10s) - 2r.So, Area = (1/2)|10s - 2r - 130|.Yes, that's correct.Set equal to 50:(1/2)|10s - 2r - 130| = 50 => |10s - 2r - 130| = 100.So, 10s - 2r - 130 = 100 or 10s - 2r - 130 = -100.So, 10s - 2r = 230 or 10s - 2r = 30.Divide both by 2: 5s - r = 115 or 5s - r = 15.So, that's correct.Then, substitute s = -3r + 61 into both equations.First equation: 5s - r = 1155*(-3r + 61) - r = 115-15r + 305 - r = 115-16r + 305 = 115-16r = 115 - 305 = -190r = (-190)/(-16) = 190/16 = 95/8 = 11.875.Then, s = -3*(95/8) + 61 = -285/8 + 488/8 = 203/8 = 25.375.So, r + s = 95/8 + 203/8 = 298/8 = 149/4 = 37.25.Second equation: 5s - r = 15Substitute s = -3r + 61:5*(-3r + 61) - r = 15-15r + 305 - r = 15-16r + 305 = 15-16r = 15 - 305 = -290r = (-290)/(-16) = 290/16 = 145/8 = 18.125.Then, s = -3*(145/8) + 61 = -435/8 + 488/8 = 53/8 = 6.625.So, r + s = 145/8 + 53/8 = 198/8 = 99/4 = 24.75.So, we have two possible solutions: (r, s) = (95/8, 203/8) with r + s = 149/4 = 37.25, and (r, s) = (145/8, 53/8) with r + s = 99/4 = 24.75.Since the problem asks for the highest possible value of r + s, it's 37.25.Wait, but in the initial assistant's solution, they got approximately 29.26, which is lower. So, why the discrepancy?Looking back, the initial assistant's solution might have made a mistake in the area equation.Wait, let me check their steps.They wrote:50 = (1/2)|r(17-15) + 10(s-17) + 20(15-s)|Which is:50 = (1/2)|2r + 10(s - 17) + 20(15 - s)|Wait, let me compute that:r(17 - 15) = 2r10(s - 17) = 10s - 17020(15 - s) = 300 - 20sSo, adding them together:2r + 10s - 170 + 300 - 20s = 2r - 10s + 130.So, Area = (1/2)|2r - 10s + 130|.Set equal to 50:(1/2)|2r - 10s + 130| = 50 => |2r - 10s + 130| = 100.So, 2r - 10s + 130 = 100 or 2r - 10s + 130 = -100.Simplify:1) 2r -10s = -30 => r -5s = -152) 2r -10s = -230 => r -5s = -115Wait, so in the initial assistant's solution, they had:100 = |2r - 30s + 470|But according to my calculation, it should be |2r -10s + 130| = 100.So, the initial assistant made a mistake in the area equation.They incorrectly expanded the terms. Instead of 10(s -17) = 10s -170 and 20(15 - s) = 300 -20s, they might have miscalculated.So, that's why their result was different.Therefore, my calculation is correct, leading to r + s = 37.25.But let me double-check.Wait, in my calculation, I used the shoelace formula correctly, leading to |10s -2r -130| = 100.Then, solving 10s -2r = 230 or 10s -2r = 30, which simplifies to 5s - r = 115 or 5s - r = 15.Substituting s = -3r +61 into both, I got r = 95/8 and r = 145/8, leading to r + s = 37.25 and 24.75.So, 37.25 is the higher value.Therefore, the highest possible value of r + s is 37.25, which is 149/4.But let me confirm if both points are valid, meaning they don't make the area negative or something.Wait, the area is absolute value, so both solutions are valid, just on opposite sides of the line.Therefore, the maximum r + s is 149/4, which is 37.25.So, the final answer is boxed{dfrac{149}{4}}.Wait, but 149/4 is 37.25, which is correct.Alternatively, as a fraction, 149/4 is already in simplest form.So, the highest possible value of r + s is 149/4.