answer：First, I need to multiply the two complex numbers ((1 + 2i)) and ((a + i)). I'll use the distributive property to expand the product.Multiplying (1) by (a) gives (a), and multiplying (1) by (i) gives (i). Next, multiplying (2i) by (a) gives (2ai), and multiplying (2i) by (i) gives (2i^2). Now, I'll combine all these terms:[a + i + 2ai + 2i^2]Since (i^2 = -1), the term (2i^2) simplifies to (-2). So, the expression becomes:[a - 2 + i + 2ai]I can group the real parts and the imaginary parts separately:[(a - 2) + (1 + 2a)i]According to the problem, the real part and the imaginary part are equal. Therefore, I set them equal to each other:[a - 2 = 1 + 2a]To solve for (a), I'll subtract (a) from both sides:[-2 = 1 + a]Then, I'll subtract (1) from both sides to isolate (a):[-3 = a]So, the value of (a) that satisfies the given condition is (-3).

answer：Alright, so I have this problem here: "Move the starting points of all vectors in the same plane, which are not less than 1 and not greater than 2, to the same point O. Then, the area of the shape formed by the endpoints of these vectors is equal to ____." Hmm, okay, let me try to understand what this is asking.First, I need to visualize what's happening. We have a bunch of vectors in the same plane. Each of these vectors has a starting point, and their lengths are between 1 and 2 units, inclusive. The problem says to move all these starting points to the same point O. So, essentially, we're translating all these vectors so that they all originate from point O instead of their original starting points.When we do this, the endpoints of these vectors will form some sort of shape. The question is asking for the area of that shape. Okay, so I need to figure out what shape is formed by all these endpoints and then calculate its area.Let me break it down step by step. First, what does it mean to move the starting points of all vectors to point O? Well, if all vectors are moved so that their starting point is at O, then their endpoints will be located at various positions around O, depending on their direction and magnitude. Since the vectors can have any direction in the plane, their endpoints will cover a region around O.Given that the vectors have lengths between 1 and 2, the endpoints can't be closer than 1 unit or farther than 2 units from O. So, intuitively, the shape formed by all these endpoints should be a ring-shaped region, right? Because all points are at least 1 unit away and at most 2 units away from O.Wait, that sounds like an annulus. An annulus is the region between two concentric circles. So, in this case, the inner circle would have a radius of 1, and the outer circle would have a radius of 2. Therefore, the shape formed by the endpoints is indeed an annulus.Now, to find the area of this annulus, I need to calculate the area of the outer circle and subtract the area of the inner circle. The formula for the area of a circle is πr², where r is the radius.So, the area of the outer circle with radius 2 is π*(2)² = 4π. The area of the inner circle with radius 1 is π*(1)² = π. Therefore, the area of the annulus is 4π - π = 3π.Wait, let me make sure I didn't make any mistakes here. The vectors can have any direction, so their endpoints should cover the entire area between the two circles, right? There are no gaps or overlaps because the vectors can point in any direction. So, yes, the shape is a complete annulus with inner radius 1 and outer radius 2.Is there another way to think about this? Maybe by considering the set of all possible endpoints. Since each vector has a magnitude between 1 and 2, and direction can be anything, the set of all endpoints is exactly the set of points in the plane that are at least 1 unit away and at most 2 units away from O. That's the definition of an annulus.So, calculating the area as 3π seems correct. I don't think I missed anything here. The key was recognizing that moving all starting points to O results in an annulus, and then applying the formula for the area of an annulus.Just to recap: Moving all vectors to start at O means their endpoints form a region around O. The lengths of the vectors determine the inner and outer boundaries of this region. Since the lengths are between 1 and 2, the region is an annulus with radii 1 and 2. The area is the difference between the areas of the two circles, which is 4π - π = 3π.I think that's solid. I don't see any errors in this reasoning. It all checks out.

answer：First, calculate the total travel time in hours over the initial period:The traveler traveled for 29 days, spending 7 hours each day traveling:[text{Total hours} = 29 text{ days} times 7 text{ hours/day} = 203 text{ hours}]Next, find the average speed of the traveler:The traveler covered 112 kilometers in 203 hours. So, the average speed ((v)) is:[v = frac{text{Total distance}}{text{Total time}} = frac{112 text{ km}}{203 text{ hours}} approx 0.5517 text{ km/hour}]Calculate the total travel time for the new period:The new period consists of 17 days, with 10 hours of travel each day:[text{Total hours} = 17 text{ days} times 10 text{ hours/day} = 170 text{ hours}]Determine the total distance traveled in the new period:Using the average speed found earlier, we calculate the total distance ((d)) as follows:[d = v times text{Total hours} = 0.5517 text{ km/hour} times 170 text{ hours} approx 93.79 text{ km}]For an exact expression, using fractions:[d = left(frac{112}{203}right) times 170 = frac{112 times 170}{203} = frac{19040}{203} approx 93.79 text{ km}]Conclusion:Therefore, the traveler will cover approximately (93 frac{23}{29}) kilometers in 17 days, traveling 10 hours per day.[boxed{93 frac{23}{29} text{ km}}]

answer：Okay, so I have this problem where a point (2, 2, -1) is being transformed through a series of operations: rotations and reflections. I need to figure out where the point ends up after all these transformations. Let me try to break this down step by step.First, the point is rotated 90 degrees about the z-axis. I remember that rotating a point around the z-axis affects the x and y coordinates. Specifically, a 90-degree rotation about the z-axis can be represented by a rotation matrix. I think the rotation matrix for 90 degrees is:[begin{pmatrix}0 & -1 & 0 1 & 0 & 0 0 & 0 & 1end{pmatrix}]So, if I apply this matrix to the point (2, 2, -1), I can calculate the new coordinates. Let me do that:- The new x-coordinate is 0*2 + (-1)*2 + 0*(-1) = 0 - 2 + 0 = -2- The new y-coordinate is 1*2 + 0*2 + 0*(-1) = 2 + 0 + 0 = 2- The z-coordinate remains the same, so it's still -1So after the first rotation, the point becomes (-2, 2, -1). Okay, that makes sense.Next, the point is reflected through the xy-plane. Reflecting through the xy-plane means that the z-coordinate changes sign. So, if the z-coordinate is -1, reflecting it would make it 1. Therefore, the point becomes (-2, 2, 1).Now, the point is reflected through the yz-plane. Reflecting through the yz-plane affects the x-coordinate, changing its sign. So, the x-coordinate is -2, and reflecting it would make it 2. So, the point becomes (2, 2, 1).Then, the point is rotated another 90 degrees about the z-axis. I need to apply the same rotation matrix again. Let's see:- The new x-coordinate is 0*2 + (-1)*2 + 0*1 = 0 - 2 + 0 = -2- The new y-coordinate is 1*2 + 0*2 + 0*1 = 2 + 0 + 0 = 2- The z-coordinate remains 1So after this rotation, the point becomes (-2, 2, 1).Finally, the point is reflected through the xy-plane again. As before, this changes the sign of the z-coordinate. So, the z-coordinate goes from 1 to -1. Therefore, the final coordinates of the point are (-2, 2, -1).Wait, let me double-check each step to make sure I didn't make a mistake.1. **First Rotation (90° about z-axis):** Applied the rotation matrix correctly, got (-2, 2, -1). That seems right.2. **First Reflection (through xy-plane):** Changed z from -1 to 1, resulting in (-2, 2, 1). Correct.3. **Second Reflection (through yz-plane):** Changed x from -2 to 2, resulting in (2, 2, 1). Correct.4. **Second Rotation (90° about z-axis):** Applied the rotation matrix again, got (-2, 2, 1). Correct.5. **Final Reflection (through xy-plane):** Changed z from 1 to -1, resulting in (-2, 2, -1). Correct.Hmm, everything seems to check out. I don't see any errors in my calculations or reasoning. So, the final coordinates should indeed be (-2, 2, -1).

answer：Okay, so I have this equation to solve: [frac{s^2 - 3s + 2}{s^2 - 6s + 5} = frac{s^2 - 4s - 5}{s^2 - 2s - 15}.] Hmm, it's a proportion with quadratic expressions in both the numerators and denominators. I remember that when dealing with equations like this, factoring is usually a good first step because it can simplify things and maybe even cancel out some terms.Let me start by factoring each quadratic expression. I'll take them one by one.First, the numerator on the left side: (s^2 - 3s + 2). To factor this, I need two numbers that multiply to 2 and add up to -3. Hmm, -1 and -2. So, it factors as ((s - 1)(s - 2)).Next, the denominator on the left side: (s^2 - 6s + 5). Looking for two numbers that multiply to 5 and add up to -6. That would be -1 and -5. So, it factors as ((s - 1)(s - 5)).Now, moving to the right side. The numerator there is (s^2 - 4s - 5). I need two numbers that multiply to -5 and add up to -4. That would be -5 and +1. So, it factors as ((s - 5)(s + 1)).Lastly, the denominator on the right side: (s^2 - 2s - 15). Looking for two numbers that multiply to -15 and add up to -2. That would be -5 and +3. So, it factors as ((s - 5)(s + 3)).Alright, so substituting all these factored forms back into the equation, it becomes:[frac{(s - 1)(s - 2)}{(s - 1)(s - 5)} = frac{(s - 5)(s + 1)}{(s - 5)(s + 3)}.]Looking at this, I notice that both sides have common factors in the numerator and denominator that can be canceled out. On the left side, ((s - 1)) appears in both the numerator and the denominator, so they cancel each other out. Similarly, on the right side, ((s - 5)) appears in both the numerator and the denominator, so they also cancel out. After canceling, the equation simplifies to:[frac{s - 2}{s - 5} = frac{s + 1}{s + 3}.]Now, I have a simpler proportion to solve. To solve for (s), I can cross-multiply. That means I'll multiply the numerator of the left side by the denominator of the right side and set it equal to the numerator of the right side multiplied by the denominator of the left side.So, cross-multiplying gives:[(s - 2)(s + 3) = (s + 1)(s - 5).]Now, I need to expand both sides of this equation. Let's start with the left side:[(s - 2)(s + 3) = s cdot s + s cdot 3 - 2 cdot s - 2 cdot 3 = s^2 + 3s - 2s - 6 = s^2 + s - 6.]Now, the right side:[(s + 1)(s - 5) = s cdot s + s cdot (-5) + 1 cdot s + 1 cdot (-5) = s^2 - 5s + s - 5 = s^2 - 4s - 5.]So, after expanding both sides, the equation becomes:[s^2 + s - 6 = s^2 - 4s - 5.]Hmm, now I can subtract (s^2) from both sides to simplify:[s^2 + s - 6 - s^2 = s^2 - 4s - 5 - s^2.]This simplifies to:[s - 6 = -4s - 5.]Now, I need to solve for (s). Let's get all the (s) terms on one side and the constants on the other. I'll add (4s) to both sides:[s - 6 + 4s = -4s - 5 + 4s.]This simplifies to:[5s - 6 = -5.]Next, I'll add 6 to both sides to isolate the term with (s):[5s - 6 + 6 = -5 + 6.]Which simplifies to:[5s = 1.]Finally, I'll divide both sides by 5 to solve for (s):[s = frac{1}{5}.]So, the solution is (s = frac{1}{5}). But wait, I should check if this value makes any denominators zero in the original equation. Let me plug (s = frac{1}{5}) back into the denominators:For the left side denominator: (s^2 - 6s + 5). Plugging in (s = frac{1}{5}):[left(frac{1}{5}right)^2 - 6 cdot frac{1}{5} + 5 = frac{1}{25} - frac{6}{5} + 5.]Calculating each term:[frac{1}{25} = 0.04,][-frac{6}{5} = -1.2,][5 = 5.]Adding them up: (0.04 - 1.2 + 5 = 3.84), which is not zero.For the right side denominator: (s^2 - 2s - 15). Plugging in (s = frac{1}{5}):[left(frac{1}{5}right)^2 - 2 cdot frac{1}{5} - 15 = frac{1}{25} - frac{2}{5} - 15.]Calculating each term:[frac{1}{25} = 0.04,][-frac{2}{5} = -0.4,][-15 = -15.]Adding them up: (0.04 - 0.4 - 15 = -15.36), which is also not zero.Therefore, (s = frac{1}{5}) is a valid solution and doesn't cause any division by zero in the original equation. So, I think that's the correct answer.**Final Answer**The value of (s) is boxed{dfrac{1}{5}}.

answer：First, I need to determine the total time Mckenna spends at work. She starts her day at 8:00 a.m. and stays until 3:00 p.m., which is a total of 7 hours.Next, I'll break down her schedule:- She works in her office from 8:00 a.m. to 11:00 a.m., which is 3 hours.- Then, she joins her team in the conference room from 11:00 a.m. to 1:00 p.m., which is another 2 hours.Adding these together, she spends 5 hours before the meeting with her team.Since her total workday is 7 hours, the time she spends working after the meeting is:7 hours (total) - 5 hours (before meeting) = 2 hours.Therefore, Mckenna works for 2 hours after the meeting with her team.