💡First, calculate John's total weekly earnings before the raises:From job A: 60 per week.From job B: 100 per week.Total earnings before raises = 60 + 100 = 160 per week.Next, calculate John's total weekly earnings after all the raises:From job A, after the raise: 78 per week.From job B, after both raises: 120 per week.Total earnings after raises = 78 + 120 = 198 per week.Now, determine the increase in earnings:Increase in earnings = Total earnings after raises - Total earnings before raisesIncrease in earnings = 198 - 160 = 38Finally, calculate the overall percentage increase:Percentage Increase = (Increase in earnings / Total earnings before raises) * 100Percentage Increase = (38 / 160) * 100Percentage Increase = 0.2375 * 100Percentage Increase = 23.75%So, the overall percentage increase in John's weekly earnings from both jobs combined is 23.75%.

💡Okay, so I have this problem with two parts about polar coordinates. Let me try to figure it out step by step.First, part (1) asks for the polar coordinates of the intersection points of curves C₁ and C₂. C₁ is given by ρ = 4 cos θ, where θ is between 0 and π/2. C₂ is given by ρ cos θ = 3. Hmm, okay. So I need to find where these two equations are true at the same time.Let me write down the equations:1. ρ = 4 cos θ2. ρ cos θ = 3Since both equations involve ρ and θ, maybe I can substitute one into the other. If I take the first equation and plug ρ into the second equation, that should give me an equation in terms of θ only.So substituting ρ from equation 1 into equation 2:(4 cos θ) * cos θ = 3Simplify that:4 cos² θ = 3Divide both sides by 4:cos² θ = 3/4Take the square root of both sides:cos θ = ±√(3)/2But wait, θ is between 0 and π/2, so cos θ is positive in that interval. So we can ignore the negative solution.So cos θ = √3 / 2What angle θ has cosine √3 / 2? I remember that cos(π/6) = √3 / 2. So θ = π/6.Now, plug θ = π/6 back into equation 1 to find ρ:ρ = 4 cos(π/6) = 4*(√3 / 2) = 2√3So the intersection point is at (2√3, π/6). That should be the answer for part (1). Let me just double-check.If I plug θ = π/6 into C₂: ρ cos θ = 3. So ρ = 3 / cos(π/6) = 3 / (√3 / 2) = 3 * 2 / √3 = 6 / √3 = 2√3. Yep, same result. So that seems correct.Okay, moving on to part (2). It says: Given point Q on C₁ with vector OQ = (2/3) vector QP, find the polar coordinate equation of the moving point P.Hmm, vectors. So OQ is the vector from the origin to Q, and QP is the vector from Q to P. The equation says that OQ is two-thirds of QP.Let me try to visualize this. If OQ = (2/3) QP, then QP is (3/2) OQ. So P is a point such that Q is between O and P, and OQ is two-thirds the length of QP.Wait, actually, let me think in terms of vectors. The vector equation is:OQ = (2/3) QPBut QP is the vector from Q to P, which is equal to OP - OQ. So QP = OP - OQ.So substituting that into the equation:OQ = (2/3)(OP - OQ)Let me write that out:OQ = (2/3) OP - (2/3) OQBring the (2/3) OQ term to the left side:OQ + (2/3) OQ = (2/3) OPCombine like terms:(1 + 2/3) OQ = (2/3) OPWhich is:(5/3) OQ = (2/3) OPMultiply both sides by 3 to eliminate denominators:5 OQ = 2 OPSo OP = (5/2) OQThat means the vector from the origin to P is (5/2) times the vector from the origin to Q. So P is just a scaled version of Q, scaled by 5/2.So if Q is a point on C₁, which is ρ = 4 cos θ, then P must be a point such that OP = (5/2) OQ.Let me denote Q in polar coordinates as (ρ₀, θ₀). Then, since Q is on C₁, ρ₀ = 4 cos θ₀.Then, P would be ( (5/2) ρ₀, θ₀ ). Because scaling the vector just scales the radius, and the angle remains the same.So substituting ρ₀ = 4 cos θ₀ into P's coordinates:ρ = (5/2) * 4 cos θ₀ = 10 cos θ₀And θ = θ₀So in terms of P's coordinates (ρ, θ), we have:ρ = 10 cos θAnd since θ₀ is between 0 and π/2, θ is also between 0 and π/2.Therefore, the polar equation for P is ρ = 10 cos θ, with θ in [0, π/2).Let me just verify this. If Q is on C₁, then Q is (4 cos θ, θ). Then P is (5/2 * 4 cos θ, θ) = (10 cos θ, θ). So yes, that makes sense. And the condition OQ = (2/3) QP is satisfied because QP = OP - OQ = (10 cos θ, θ) - (4 cos θ, θ) = (6 cos θ, θ). So OQ = (4 cos θ, θ) = (2/3)(6 cos θ, θ) = (2/3) QP. Yep, that checks out.So I think that's the correct answer for part (2).**Final Answer**(1) The polar coordinates of the intersection point are boxed{(2sqrt{3}, frac{pi}{6})}.(2) The polar coordinate equation of the moving point ( P ) is boxed{rho = 10costheta} for ( 0 leq theta < frac{pi}{2} ).

💡Okay, so I have this problem about a parabola and its reflection. Let me try to understand what it's asking. The original parabola is given by the equation y = ax² + bx + c. Then, it's reflected over the x-axis. After that, the original parabola is moved 3 units to the right, and its reflection is moved 4 units to the left. I need to find the expression for y = (f + g)(x), where f(x) is the equation of the translated original parabola and g(x) is the equation of the translated reflection.Alright, let's break this down step by step. First, reflecting a parabola over the x-axis. I remember that reflecting a graph over the x-axis changes the sign of the y-values. So, if the original equation is y = ax² + bx + c, then the reflection over the x-axis should be y = -ax² - bx - c. That makes sense because reflecting over the x-axis flips the graph vertically.Next, translating the original parabola 3 units to the right. Translating a function horizontally involves shifting its graph left or right. For a function f(x), shifting it h units to the right is done by replacing x with (x - h). So, in this case, replacing x with (x - 3) in the original equation. Let me write that out:Original parabola after translation: y = a(x - 3)² + b(x - 3) + c.I need to expand this to make it easier to work with later. Let's do that step by step.First, expand (x - 3)²: that's x² - 6x + 9. So, multiplying by a gives a(x² - 6x + 9) = ax² - 6ax + 9a.Next, expand b(x - 3): that's bx - 3b.Adding the constant term c.So, putting it all together:y = ax² - 6ax + 9a + bx - 3b + c.Combine like terms:The x² term is ax².The x terms are (-6a + b)x.The constant terms are 9a - 3b + c.So, the translated original parabola is y = ax² + (-6a + b)x + (9a - 3b + c).Alright, that's f(x).Now, let's work on the reflection. The reflection was y = -ax² - bx - c. Now, we need to translate this reflection 4 units to the left. Translating a function h units to the left is done by replacing x with (x + h). So, replacing x with (x + 4) in the reflection equation.So, the reflected parabola after translation: y = -a(x + 4)² - b(x + 4) - c.Again, let's expand this step by step.First, expand (x + 4)²: that's x² + 8x + 16. Multiplying by -a gives -a(x² + 8x + 16) = -ax² - 8ax - 16a.Next, expand -b(x + 4): that's -bx - 4b.Subtracting c.Putting it all together:y = -ax² - 8ax - 16a - bx - 4b - c.Combine like terms:The x² term is -ax².The x terms are (-8a - b)x.The constant terms are -16a - 4b - c.So, the translated reflected parabola is y = -ax² + (-8a - b)x + (-16a - 4b - c).Alright, that's g(x).Now, I need to find (f + g)(x), which means adding f(x) and g(x) together.Let's write down f(x) and g(x):f(x) = ax² + (-6a + b)x + (9a - 3b + c)g(x) = -ax² + (-8a - b)x + (-16a - 4b - c)Adding these together term by term.First, the x² terms: ax² + (-ax²) = 0. They cancel each other out.Next, the x terms: (-6a + b)x + (-8a - b)x.Let's combine the coefficients:-6a + b -8a - b = (-6a -8a) + (b - b) = -14a + 0 = -14a.So, the x term is -14a x.Now, the constant terms: (9a - 3b + c) + (-16a - 4b - c).Combine like terms:9a -16a = -7a-3b -4b = -7bc - c = 0So, the constant term is -7a -7b.Putting it all together, (f + g)(x) = -14a x -7a -7b.Hmm, that's interesting. So, the resulting function is a linear function because the x² terms canceled out. It's a straight line with a slope of -14a and a y-intercept of -7a -7b.Wait, let me double-check my calculations to make sure I didn't make a mistake.Starting with f(x):Original parabola translated 3 units right: y = a(x - 3)² + b(x - 3) + c.Expanding:a(x² -6x +9) + b(x -3) + c = ax² -6ax +9a +bx -3b +c.Combine like terms:ax² + (-6a + b)x + (9a -3b +c). That looks correct.g(x):Reflected parabola translated 4 units left: y = -a(x +4)² -b(x +4) -c.Expanding:-a(x² +8x +16) -bx -4b -c = -ax² -8ax -16a -bx -4b -c.Combine like terms:-ax² + (-8a -b)x + (-16a -4b -c). That also looks correct.Adding f(x) and g(x):x² terms: ax² -ax² = 0.x terms: (-6a + b)x + (-8a -b)x = (-6a -8a) + (b -b) = -14a x.Constants: (9a -3b +c) + (-16a -4b -c) = (9a -16a) + (-3b -4b) + (c -c) = -7a -7b.Yes, that seems correct. So, the sum is y = -14a x -7a -7b.Which is a linear equation, meaning it's a straight line. Since the coefficient of x is -14a, unless a is zero, which it isn't because it's a parabola, the slope is non-zero. Therefore, it's a non-horizontal line.So, the expression for y = (f + g)(x) is y = -14a x -7a -7b, which is a non-horizontal line.I think that's the answer. Let me just make sure I didn't miss anything in the problem statement.The original parabola is reflected over the x-axis, then the original is translated right 3, reflection translated left 4. Then sum f and g. Yes, that's what I did.I think I'm confident with this result.

💡Okay, so I have this problem where Cookie Monster encounters a cookie shaped region that's the intersection of a circle and an ellipse. The circle is given by the equation ((x-2)^2 + (y-1)^2 = 16), and the ellipse is given by (frac{(x-2)^2}{25} + frac{(y-1)^2}{9} = 1). I need to determine the radius of the circle.Alright, let me start by recalling what the standard equations of circles and ellipses look like. For a circle, the standard form is ((x - h)^2 + (y - k)^2 = r^2), where ((h, k)) is the center and (r) is the radius. For an ellipse, the standard form is (frac{(x - h)^2}{a^2} + frac{(y - k)^2}{b^2} = 1), where ((h, k)) is the center, (a) is the semi-major axis, and (b) is the semi-minor axis.Looking at the circle's equation ((x-2)^2 + (y-1)^2 = 16), I can see that the center of the circle is at ((2, 1)). The right side of the equation is (16), which is (r^2), so to find the radius (r), I take the square root of 16. That gives me (r = 4). So, the radius of the circle is 4 units.Now, just to make sure I'm not missing anything, I'll check the ellipse's equation too. The ellipse is given by (frac{(x-2)^2}{25} + frac{(y-1)^2}{9} = 1). Here, the center is also at ((2, 1)), which means both the circle and the ellipse are centered at the same point. The denominators under the squared terms give me the squares of the semi-major and semi-minor axes. So, the semi-major axis is (sqrt{25} = 5) and the semi-minor axis is (sqrt{9} = 3). That tells me the ellipse is stretched out more along the x-axis compared to the y-axis.But wait, the problem is asking for the radius of the circle, not the ellipse. So, even though the ellipse is part of the problem, I don't need to calculate anything related to it for this specific question. The radius is solely determined by the circle's equation.Just to double-check, I'll visualize both shapes. The circle is centered at (2,1) with a radius of 4, so it extends 4 units in all directions from the center. The ellipse, on the other hand, extends 5 units along the x-axis and 3 units along the y-axis from the same center. The intersection of these two shapes would form the cookie region, but since the question only asks for the radius of the circle, I don't need to delve deeper into their intersection.I think I've got it. The radius is clearly 4, as derived from the circle's equation. There's no ambiguity here because the equation is in the standard form, making it straightforward to identify the radius.**Final Answer**The radius of the circle is boxed{4}.

💡First, I need to determine the total sales amount that would result in a total commission of Rs. 1380.The salesman earns a 9% commission on the total sales. If the sales exceed Rs. 10,000, he also receives an additional 3% commission on the amount exceeding Rs. 10,000.Let’s denote the total sales as ( S ).The commission on the total sales is ( 0.09S ).The additional commission on the excess amount over Rs. 10,000 is ( 0.03(S - 10,000) ).The total commission is the sum of these two amounts:[0.09S + 0.03(S - 10,000) = 1380]Simplifying the equation:[0.09S + 0.03S - 300 = 1380][0.12S = 1680][S = frac{1680}{0.12} = 14,000]Now that I have the total sales (( S = 14,000 )), I can calculate the bonus.The excess amount over Rs. 10,000 is:[14,000 - 10,000 = 4,000]The bonus is 3% of this excess amount:[0.03 times 4,000 = 120]

💡Alright, so I have this problem here about two propositions, A and B. Proposition A says, "If a real number ( a ) satisfies ( a < 2 )." Proposition B is, "( a^2 < 4 )." The question is asking me to figure out what kind of condition Proposition A is for Proposition B. The options are necessary, sufficient, necessary but not sufficient, or not necessary and not sufficient.Hmm, okay. Let me break this down. First, I need to understand what each proposition is saying. Proposition A is a statement about a real number ( a ) being less than 2. Proposition B is a statement about the square of ( a ) being less than 4. So, I need to see how these two statements relate to each other in terms of logical conditions.I remember that in logic, a necessary condition is something that must be true for another statement to be true, but it doesn't guarantee it. A sufficient condition, on the other hand, is something that, if it's true, guarantees the truth of another statement, but it's not necessarily the only way for that statement to be true.So, in this case, I need to see if Proposition A is necessary for Proposition B, sufficient, both, or neither. Let me think about the relationship between ( a < 2 ) and ( a^2 < 4 ).First, let's analyze Proposition B: ( a^2 < 4 ). I know that ( a^2 < 4 ) implies that ( a ) is between -2 and 2. So, mathematically, ( a^2 < 4 ) is equivalent to ( -2 < a < 2 ). That makes sense because squaring a number between -2 and 2 will give a result less than 4.Now, looking at Proposition A: ( a < 2 ). This means that ( a ) can be any real number less than 2, which includes numbers less than -2 as well. For example, ( a = -3 ) satisfies ( a < 2 ), but ( (-3)^2 = 9 ), which is not less than 4. So, just knowing that ( a < 2 ) doesn't necessarily mean that ( a^2 < 4 ). Therefore, Proposition A isn't a sufficient condition for Proposition B because it doesn't guarantee that ( a^2 < 4 ).But is Proposition A a necessary condition? Let's see. If ( a^2 < 4 ), then ( a ) must be between -2 and 2, which means ( a < 2 ) is definitely true. So, for ( a^2 < 4 ) to hold, ( a < 2 ) must hold. Therefore, Proposition A is a necessary condition for Proposition B because without ( a < 2 ), ( a^2 < 4 ) can't be true.Putting this together, Proposition A is necessary because ( a^2 < 4 ) requires ( a < 2 ), but it's not sufficient because ( a < 2 ) doesn't necessarily lead to ( a^2 < 4 ). So, the correct term is that Proposition A is a necessary but not sufficient condition for Proposition B.Wait, let me double-check that. If ( a^2 < 4 ), then ( a ) must be less than 2, so ( a < 2 ) is necessary. But if ( a < 2 ), ( a ) could be less than -2, which would make ( a^2 ) greater than or equal to 4. So, yeah, it's not sufficient. That seems right.I think I've got it. So, the answer should be that Proposition A is a necessary but not sufficient condition for Proposition B.**Final Answer**boxed{text{necessary but not sufficient condition}}