💡Okay, so I have this problem where I need to find the minimum value of the expression (dfrac{1}{x+y} + dfrac{2}{y+z}) given that (x), (y), and (z) are positive real numbers and (x + 2y + z = 1). Hmm, let me try to figure this out step by step.First, I notice that the constraint is (x + 2y + z = 1). Maybe I can express some variables in terms of others to simplify the problem. Let me see... If I let (x + y = t), then since (x + 2y + z = 1), substituting (x + y = t) gives me (t + y + z = 1). Wait, that doesn't seem right. Let me try again.If (x + y = t), then from the constraint (x + 2y + z = 1), I can write (t + y + z = 1). So, (y + z = 1 - t). That makes sense because (x + y = t) and (y + z = 1 - t). So, now I can rewrite the original expression in terms of (t).The expression becomes (dfrac{1}{t} + dfrac{2}{1 - t}). Now, I need to find the minimum value of this expression with respect to (t), where (t) is between 0 and 1 because (x), (y), and (z) are positive real numbers.So, let me denote (f(t) = dfrac{1}{t} + dfrac{2}{1 - t}). To find the minimum, I can take the derivative of (f(t)) with respect to (t) and set it equal to zero.Calculating the derivative:[f'(t) = -dfrac{1}{t^2} + dfrac{2}{(1 - t)^2}]Setting (f'(t) = 0):[-dfrac{1}{t^2} + dfrac{2}{(1 - t)^2} = 0]Moving one term to the other side:[dfrac{2}{(1 - t)^2} = dfrac{1}{t^2}]Cross-multiplying:[2t^2 = (1 - t)^2]Expanding the right side:[2t^2 = 1 - 2t + t^2]Subtracting (t^2) from both sides:[t^2 = 1 - 2t]Bringing all terms to one side:[t^2 + 2t - 1 = 0]This is a quadratic equation. Using the quadratic formula:[t = dfrac{-2 pm sqrt{(2)^2 - 4(1)(-1)}}{2(1)} = dfrac{-2 pm sqrt{4 + 4}}{2} = dfrac{-2 pm sqrt{8}}{2} = dfrac{-2 pm 2sqrt{2}}{2} = -1 pm sqrt{2}]Since (t) must be between 0 and 1, we discard the negative solution:[t = -1 + sqrt{2}]So, (t = sqrt{2} - 1). Now, let's plug this back into the expression (f(t)):[f(t) = dfrac{1}{sqrt{2} - 1} + dfrac{2}{1 - (sqrt{2} - 1)} = dfrac{1}{sqrt{2} - 1} + dfrac{2}{2 - sqrt{2}}]To simplify these fractions, I'll rationalize the denominators.First term:[dfrac{1}{sqrt{2} - 1} times dfrac{sqrt{2} + 1}{sqrt{2} + 1} = dfrac{sqrt{2} + 1}{(sqrt{2})^2 - 1^2} = dfrac{sqrt{2} + 1}{2 - 1} = sqrt{2} + 1]Second term:[dfrac{2}{2 - sqrt{2}} times dfrac{2 + sqrt{2}}{2 + sqrt{2}} = dfrac{2(2 + sqrt{2})}{(2)^2 - (sqrt{2})^2} = dfrac{4 + 2sqrt{2}}{4 - 2} = dfrac{4 + 2sqrt{2}}{2} = 2 + sqrt{2}]Adding both terms together:[sqrt{2} + 1 + 2 + sqrt{2} = 3 + 2sqrt{2}]So, the minimum value of the expression is (3 + 2sqrt{2}), which corresponds to option C.Wait, let me double-check my steps to make sure I didn't make any mistakes. I set (x + y = t) and (y + z = 1 - t), which seems correct. Then I took the derivative of (f(t)) and solved for (t), getting (t = sqrt{2} - 1). Plugging that back in, I rationalized the denominators correctly and added the terms. It all seems to check out. So, I think my answer is correct.**Final Answer**boxed{C}

💡Alright, so I've got this problem about three clients at a hairdresser's, each paying their bill at the cash register. The problem states that each client pays the same amount that's in the register and takes 10 reais as change. After all three clients have made their payments, the cash register is empty. I need to find out how much money was in the cash register at the beginning.Okay, let's break this down step by step. First, I need to understand what happens with each client. Each client pays an amount equal to what's currently in the register and then takes 10 reais as change. So, if there's x amount in the register initially, the first client pays x reais and then takes back 10 reais as change. That means the net amount the register gains from the first client is x - 10 reais.Wait, hold on. If the client pays x reais and takes back 10 reais, then the register's total after the first transaction would be x (initial amount) plus (x - 10) from the client's payment. So, the new amount in the register after the first client would be x + (x - 10) = 2x - 10.Okay, that makes sense. Now, the second client does the same thing. The amount in the register is now 2x - 10. So, the second client pays 2x - 10 reais and takes back 10 reais as change. Therefore, the net gain for the register from the second client is (2x - 10) - 10 = 2x - 20. Adding that to the current amount in the register, which is 2x - 10, the new total becomes (2x - 10) + (2x - 20) = 4x - 30.Alright, moving on to the third client. The register now has 4x - 30 reais. The third client pays this entire amount, 4x - 30 reais, and takes back 10 reais as change. So, the net gain from the third client is (4x - 30) - 10 = 4x - 40. Adding this to the current amount in the register, which is 4x - 30, the new total becomes (4x - 30) + (4x - 40) = 8x - 70.According to the problem, after the third client, the register is empty. That means 8x - 70 equals zero. So, I can set up the equation:8x - 70 = 0To solve for x, I'll add 70 to both sides:8x = 70Then, divide both sides by 8:x = 70 / 8x = 8.75So, the initial amount in the cash register was 8.75 reais.Wait a minute, let me double-check that. If the initial amount was 8.75 reais, then:First client pays 8.75 reais and takes back 10 reais. So, the register would have 8.75 + (8.75 - 10) = 8.75 + (-1.25) = 7.50 reais.Second client pays 7.50 reais and takes back 10 reais. So, the register would have 7.50 + (7.50 - 10) = 7.50 + (-2.50) = 5.00 reais.Third client pays 5.00 reais and takes back 10 reais. So, the register would have 5.00 + (5.00 - 10) = 5.00 + (-5.00) = 0 reais.Okay, that checks out. So, the initial amount was indeed 8.75 reais.

💡Okay, so I have this question about car exhaust and air quality. It's asking about the relationship between the two. The options are positive correlation, negative correlation, no correlation, or uncertain. Hmm, let me think about this step by step.First, I know that car exhaust refers to the gases and particles that come out of a car's tailpipe when it's running. These include things like carbon dioxide, carbon monoxide, nitrogen oxides, and particulate matter. I've heard that these emissions can be harmful to the environment and to human health.Now, air quality refers to how clean or polluted the air is. Good air quality means the air is clean and safe to breathe, while poor air quality means there are pollutants in the air that can be harmful. So, if car exhaust is contributing to air pollution, that would mean it's making the air quality worse.Let me think about what a positive correlation means. That would imply that as one thing increases, the other also increases. So, if car exhaust increases, air quality would also increase. But wait, that doesn't make sense because more car exhaust should mean worse air quality, not better.On the other hand, a negative correlation means that as one thing increases, the other decreases. So, if car exhaust increases, air quality decreases. That makes more sense because more pollutants in the air would lead to poorer air quality.No correlation would mean that car exhaust doesn't affect air quality at all, which I don't think is true because I know car emissions are a significant source of air pollution.Uncertain means that we don't know for sure, but I think there's enough evidence to say that car exhaust does affect air quality negatively.So, putting it all together, car exhaust contributes to air pollution, which in turn decreases air quality. Therefore, there's a negative correlation between car exhaust and air quality.

💡Alright, so I have this problem about a sphere intersecting three different coordinate planes, and I need to find the radius of the circle where it intersects the yz-plane. Let me try to visualize this and break it down step by step.First, the sphere intersects the xy-plane at a circle centered at (3,5,0) with a radius of 2. Okay, so if I imagine the xy-plane, which is the horizontal plane where z=0, the sphere cuts through it creating a circle. The center of this circle is at (3,5,0), which means the sphere's center must be somewhere along the vertical line passing through (3,5,0). So, the x and y coordinates of the sphere's center are probably 3 and 5, respectively. But what about the z-coordinate?Next, the sphere also intersects the yz-plane at a circle centered at (0,5,-8). The yz-plane is the vertical plane where x=0. So, the sphere cuts through this plane, creating another circle. The center of this circle is at (0,5,-8). Hmm, so similar to the xy-plane intersection, this tells me that the y and z coordinates of the sphere's center are 5 and -8, respectively. But wait, earlier I thought the x-coordinate was 3. So, putting this together, the sphere's center must be at (3,5,-8). That makes sense because it has to satisfy both the xy-plane and yz-plane intersections.Let me confirm this by considering the third intersection: the sphere intersects the xz-plane at a circle centered at (3,0,-8). The xz-plane is where y=0. So, the center of this circle is at (3,0,-8). Again, this suggests that the x and z coordinates of the sphere's center are 3 and -8, respectively. Since we already concluded the center is (3,5,-8), this checks out because the y-coordinate here is 0, which is consistent with the xz-plane intersection.Okay, so now I'm confident that the sphere's center is at (3,5,-8). Now, I need to find the radius of the sphere. To do this, I can use the information from one of the intersections. Let's take the intersection with the xy-plane first.The circle in the xy-plane has a radius of 2. The distance from the sphere's center to this circle's center is along the z-axis. Since the sphere's center is at (3,5,-8) and the circle's center is at (3,5,0), the distance between them is the absolute difference in the z-coordinates, which is | -8 - 0 | = 8 units.Now, if I imagine a right triangle where one leg is this distance (8 units) and the other leg is the radius of the circle in the xy-plane (2 units), then the hypotenuse of this triangle would be the radius of the sphere. So, using the Pythagorean theorem:Sphere radius (R) = sqrt( (distance)^2 + (circle radius)^2 )R = sqrt(8^2 + 2^2)R = sqrt(64 + 4)R = sqrt(68)Okay, so the sphere has a radius of sqrt(68). Now, I need to find the radius of the circle where the sphere intersects the yz-plane. Let's denote this radius as r.Again, I can use the Pythagorean theorem here. The distance from the sphere's center to the center of the circle in the yz-plane is along the x-axis. The sphere's center is at (3,5,-8), and the circle's center is at (0,5,-8). So, the distance between them is |3 - 0| = 3 units.Now, considering another right triangle where one leg is this distance (3 units) and the hypotenuse is the sphere's radius (sqrt(68)), the other leg would be the radius of the circle in the yz-plane (r). So:r = sqrt( (sphere radius)^2 - (distance)^2 )r = sqrt(68 - 3^2)r = sqrt(68 - 9)r = sqrt(59)Wait, let me double-check my steps to make sure I didn't make a mistake. I found the sphere's center correctly by considering the centers of the intersections. Then, using the distance from the sphere's center to the xy-plane intersection, I calculated the sphere's radius as sqrt(68). Then, using the distance from the sphere's center to the yz-plane intersection, which is 3 units, I applied the Pythagorean theorem again to find the radius of the yz-plane intersection, which came out to sqrt(59). That seems consistent.Just to be thorough, let me check with the third intersection as well. The circle in the xz-plane is centered at (3,0,-8). The distance from the sphere's center (3,5,-8) to this circle's center is along the y-axis, which is |5 - 0| = 5 units. Using the sphere's radius sqrt(68), the radius of this circle should be:r = sqrt(68 - 5^2)r = sqrt(68 - 25)r = sqrt(43)But wait, the problem didn't ask for this radius, so maybe I don't need to worry about it. However, it's good to know that the calculations are consistent across all three intersections.So, going back, I think my process is correct. I found the sphere's center by considering the centers of the intersections, then used the Pythagorean theorem twice to find the sphere's radius and then the radius of the yz-plane intersection. Therefore, I'm confident that the radius r is sqrt(59).

💡Alright, so I have this problem about a right square pyramid V-ABCD, and all five of its vertices lie on the same spherical surface. The base edge length is 4, and the side edge length is (2sqrt{6}). I need to find the volume of this sphere. Hmm, okay, let me break this down step by step.First, let me visualize the pyramid. It's a right square pyramid, which means the base is a square, and the apex (the top vertex) is directly above the center of the base. All the vertices, including the apex and the four base vertices, lie on a sphere. So, this sphere is the circumscribed sphere around the pyramid.Given:- Base edge length ((a)) = 4- Side edge length ((e)) = (2sqrt{6})I need to find the radius of this sphere to calculate its volume. Once I have the radius, the volume is straightforward using the formula (V = frac{4}{3}pi R^3).But first, how do I find the radius of the circumscribed sphere? I think I need to find the center of the sphere and then determine the distance from this center to any of the pyramid's vertices, which will be the radius.Since it's a right square pyramid, the apex is directly above the center of the base. Let me denote the center of the base as point O. So, the apex V is somewhere along the line perpendicular to the base at point O.Let me consider the coordinates to make this easier. Maybe placing the pyramid in a coordinate system will help. Let's place the base ABCD on the xy-plane with its center O at the origin (0, 0, 0). Then, the apex V will be at some point (0, 0, h), where h is the height of the pyramid.Given that the base edge length is 4, each side of the square base is 4 units. So, the distance from the center O to any base vertex (like A, B, C, D) is half the diagonal of the base. Wait, actually, no. The distance from the center to a vertex is half the length of the diagonal of the square base.The diagonal of a square with side length (a) is (asqrt{2}). So, for a base edge length of 4, the diagonal is (4sqrt{2}). Therefore, the distance from the center O to any base vertex is half of that, which is (2sqrt{2}).So, the coordinates of the base vertices can be:- A: (2, 2, 0)- B: (-2, 2, 0)- C: (-2, -2, 0)- D: (2, -2, 0)Wait, actually, if the side length is 4, then from the center O(0,0,0), each vertex is at a distance of (2sqrt{2}). So, the coordinates should be such that each vertex is 2 units away from the center along both x and y axes. So, actually, it's (2, 2, 0), (-2, 2, 0), etc., but that would make the edge length between A and B as the distance between (2,2,0) and (-2,2,0), which is 4 units. Wait, no, that's not correct because the distance between (2,2,0) and (-2,2,0) is 4 units along the x-axis, but the edge length is supposed to be 4. Hmm, actually, that works because the edge length is the distance between two adjacent vertices, which in this case is 4 units. So, that seems correct.Now, the apex V is at (0, 0, h). The side edge length is given as (2sqrt{6}). The side edge is the distance from the apex V to any base vertex, say A. So, the distance between V(0,0,h) and A(2,2,0) is (2sqrt{6}).Let me write that as an equation:[sqrt{(2 - 0)^2 + (2 - 0)^2 + (0 - h)^2} = 2sqrt{6}]Simplifying inside the square root:[sqrt{4 + 4 + h^2} = 2sqrt{6}][sqrt{8 + h^2} = 2sqrt{6}]Squaring both sides:[8 + h^2 = (2sqrt{6})^2][8 + h^2 = 4 times 6][8 + h^2 = 24][h^2 = 24 - 8][h^2 = 16][h = 4]So, the height of the pyramid is 4 units. That means the apex V is at (0, 0, 4).Now, I need to find the radius R of the circumscribed sphere. Since all five vertices lie on the sphere, the distance from the center of the sphere to each vertex is R.But where is the center of the sphere? In a right square pyramid, the center of the circumscribed sphere isn't necessarily at the centroid of the pyramid. It might be somewhere along the axis of the pyramid, but not necessarily at the apex or the base center.Let me denote the center of the sphere as point C, located somewhere along the z-axis, say at (0, 0, k). Then, the distance from C to any vertex (base or apex) must be R.So, let's write the equations for the distances from C to a base vertex and to the apex.First, distance from C(0,0,k) to A(2,2,0):[sqrt{(2 - 0)^2 + (2 - 0)^2 + (0 - k)^2} = R][sqrt{4 + 4 + k^2} = R][sqrt{8 + k^2} = R]Second, distance from C(0,0,k) to V(0,0,4):[sqrt{(0 - 0)^2 + (0 - 0)^2 + (4 - k)^2} = R][sqrt{(4 - k)^2} = R][|4 - k| = R]Since R is positive, we can write (4 - k = R) or (k - 4 = R). But since k is the z-coordinate of the center, it's somewhere between the base and the apex. If k is above the apex, then k > 4, but that would make R = k - 4, but then the distance from C to the base vertices would be larger. Alternatively, if k is below the apex, then 4 - k is positive, so R = 4 - k.But let's see. If k is between 0 and 4, then 4 - k is positive, and R = 4 - k. If k is above 4, R = k - 4, but then the distance from C to the base vertices would be larger than R, which might not be possible because the sphere must enclose all points.Wait, actually, the sphere must enclose all points, so the center could be either above or below the pyramid, but in this case, since the pyramid is above the base, the sphere's center is likely somewhere along the z-axis above the base but below the apex.Wait, no, actually, it could be anywhere along the z-axis. Let me think.If I have the apex at (0,0,4) and the base vertices at z=0, the sphere must enclose all these points. So, the center could be somewhere between z=0 and z=4, or even outside that range, but likely between.Let me proceed with the equations.From the apex V(0,0,4):[R = |4 - k|]From the base vertex A(2,2,0):[R = sqrt{8 + k^2}]So, setting these equal:[|4 - k| = sqrt{8 + k^2}]Since we're dealing with distances, R must be positive, so we can square both sides without worrying about the absolute value:[(4 - k)^2 = 8 + k^2][16 - 8k + k^2 = 8 + k^2]Subtract (k^2) from both sides:[16 - 8k = 8][16 - 8 = 8k][8 = 8k][k = 1]So, the center of the sphere is at (0,0,1). Therefore, the radius R is:[R = |4 - k| = |4 - 1| = 3]Alternatively, plugging k=1 into the other equation:[R = sqrt{8 + (1)^2} = sqrt{9} = 3]So, R=3.Now, the volume of the sphere is:[V = frac{4}{3}pi R^3 = frac{4}{3}pi (3)^3 = frac{4}{3}pi times 27 = 36pi]So, the volume of the sphere is (36pi), which corresponds to option B.Wait, let me double-check my steps to make sure I didn't make any mistakes.1. Calculated the distance from the center of the base to a vertex: (2sqrt{2}). That seems correct because for a square with side length 4, the diagonal is (4sqrt{2}), so half of that is (2sqrt{2}).2. Set up the coordinates correctly, with the apex at (0,0,4) and base vertices at (2,2,0), etc.3. Calculated the height h using the side edge length: yes, that gave h=4, which makes sense.4. Then, set up the equations for the sphere's center at (0,0,k). That's logical because the pyramid is symmetric around the z-axis, so the sphere's center must lie on the z-axis.5. Equated the distances from the center to a base vertex and to the apex, leading to the equation (|4 - k| = sqrt{8 + k^2}). Solved for k and got k=1, so R=3.6. Calculated the volume as (36pi), which is one of the options.Everything seems to check out. I don't see any errors in the calculations or reasoning.**Final Answer**boxed{B}

💡Okay, so I have this trigonometry problem here, and I need to find the value of tan(alpha + frac{pi}{4}) given that tan(alpha + beta) = frac{2}{5} and tan(beta - frac{pi}{4}) = frac{1}{4}. Hmm, let me think about how to approach this.First, I remember that there are formulas for the tangent of sums and differences of angles. Maybe I can use those to relate the given information to the expression I need to find. The tangent addition formula is:tan(A + B) = frac{tan A + tan B}{1 - tan A tan B}And the tangent subtraction formula is:tan(A - B) = frac{tan A - tan B}{1 + tan A tan B}Looking at the problem, I have tan(alpha + beta) and tan(beta - frac{pi}{4}). I need to find tan(alpha + frac{pi}{4}). Maybe I can express alpha + frac{pi}{4} in terms of the given angles.Let me see, if I consider alpha + frac{pi}{4}, can I write this as (alpha + beta) - (beta - frac{pi}{4})? Let me check:(alpha + beta) - (beta - frac{pi}{4}) = alpha + beta - beta + frac{pi}{4} = alpha + frac{pi}{4}Yes, that works! So, alpha + frac{pi}{4} is equal to (alpha + beta) - (beta - frac{pi}{4}). Therefore, I can use the tangent subtraction formula here.Let me denote A = alpha + beta and B = beta - frac{pi}{4}. Then, tan(A - B) = tan(alpha + frac{pi}{4}).Using the tangent subtraction formula:tan(A - B) = frac{tan A - tan B}{1 + tan A tan B}Substituting the known values:tan(alpha + frac{pi}{4}) = frac{tan(alpha + beta) - tan(beta - frac{pi}{4})}{1 + tan(alpha + beta)tan(beta - frac{pi}{4})}Plugging in the given values:tan(alpha + frac{pi}{4}) = frac{frac{2}{5} - frac{1}{4}}{1 + frac{2}{5} times frac{1}{4}}Now, let's compute the numerator and the denominator separately.First, the numerator:frac{2}{5} - frac{1}{4} = frac{8}{20} - frac{5}{20} = frac{3}{20}Wait, is that right? Let me double-check:frac{2}{5} is equal to frac{8}{20}, and frac{1}{4} is equal to frac{5}{20}. Subtracting them gives frac{8}{20} - frac{5}{20} = frac{3}{20}. Okay, that seems correct.Now, the denominator:1 + frac{2}{5} times frac{1}{4} = 1 + frac{2}{20} = 1 + frac{1}{10} = frac{10}{10} + frac{1}{10} = frac{11}{10}Wait, let me verify that:frac{2}{5} times frac{1}{4} = frac{2}{20} = frac{1}{10}. Adding 1 gives 1 + frac{1}{10} = frac{11}{10}. That's correct.So now, putting it all together:tan(alpha + frac{pi}{4}) = frac{frac{3}{20}}{frac{11}{10}} = frac{3}{20} times frac{10}{11} = frac{30}{220} = frac{3}{22}Simplifying frac{30}{220} by dividing numerator and denominator by 10 gives frac{3}{22}. So, the value of tan(alpha + frac{pi}{4}) is frac{3}{22}.Let me just make sure I didn't make any calculation errors. Starting from the beginning:1. Expressed alpha + frac{pi}{4} as (alpha + beta) - (beta - frac{pi}{4}).2. Applied the tangent subtraction formula.3. Substituted the given tangents: frac{2}{5} and frac{1}{4}.4. Calculated the numerator: frac{2}{5} - frac{1}{4} = frac{8}{20} - frac{5}{20} = frac{3}{20}.5. Calculated the denominator: 1 + frac{2}{5} times frac{1}{4} = 1 + frac{1}{10} = frac{11}{10}.6. Divided the numerator by the denominator: frac{3}{20} div frac{11}{10} = frac{3}{20} times frac{10}{11} = frac{3}{22}.Everything seems to check out. So, the answer should be frac{3}{22}, which corresponds to option C.**Final Answer**boxed{C}