💡First, I need to determine the percentage of solution P in the mixture. I'll start by defining the volumes of solutions P and Q as ( x ) and ( y ) respectively.Next, I'll set up an equation based on the percentage of carbonated water in the mixture. Solution P contains 80% carbonated water, and solution Q contains 55% carbonated water. The mixture has 67.5% carbonated water.This gives me the equation:[0.80x + 0.55y = 0.675(x + y)]I'll simplify this equation to find the relationship between ( x ) and ( y ):[0.80x + 0.55y = 0.675x + 0.675y][0.125x = 0.125y][x = y]This means the volumes of solutions P and Q are equal. Therefore, the percentage of solution P in the mixture is:[frac{x}{x + y} times 100% = frac{x}{2x} times 100% = 50%]

💡Okay, so I have this problem here where I'm given parametric equations for a curve, and I need to find its Cartesian equation. The parametric equations are:x = 1 + 2 cos θy = 2 + 3 sin θAnd θ is the parameter. The options are all different equations, some with pluses and some with minuses, so I need to figure out which one is correct.First, I remember that parametric equations can often be converted to Cartesian form by eliminating the parameter, which in this case is θ. Since both x and y are expressed in terms of cos θ and sin θ, maybe I can use the Pythagorean identity, which is cos²θ + sin²θ = 1. That seems like a good starting point.So, let me try to express cos θ and sin θ in terms of x and y. From the first equation, x = 1 + 2 cos θ, I can solve for cos θ. Subtracting 1 from both sides gives x - 1 = 2 cos θ, and then dividing both sides by 2, I get cos θ = (x - 1)/2.Similarly, from the second equation, y = 2 + 3 sin θ, I can solve for sin θ. Subtracting 2 from both sides gives y - 2 = 3 sin θ, and dividing both sides by 3, I get sin θ = (y - 2)/3.Now that I have expressions for both cos θ and sin θ in terms of x and y, I can plug them into the Pythagorean identity. So, cos²θ + sin²θ = 1 becomes [(x - 1)/2]^2 + [(y - 2)/3]^2 = 1.Let me write that out:[(x - 1)/2]^2 + [(y - 2)/3]^2 = 1Simplifying each term, squaring the numerators and denominators:(x - 1)^2 / 4 + (y - 2)^2 / 9 = 1Hmm, that looks like the equation of an ellipse. The standard form of an ellipse is (x - h)^2 / a^2 + (y - k)^2 / b^2 = 1, where (h, k) is the center, and a and b are the lengths of the semi-major and semi-minor axes.In this case, the center of the ellipse would be at (1, 2), which makes sense because when θ = 0, cos θ = 1 and sin θ = 0, so x = 1 + 2(1) = 3 and y = 2 + 3(0) = 2. Similarly, when θ = π/2, cos θ = 0 and sin θ = 1, so x = 1 + 2(0) = 1 and y = 2 + 3(1) = 5. So, the ellipse is centered at (1, 2) and extends 2 units to the left and right along the x-axis and 3 units up and down along the y-axis.Looking back at the options provided:A) [(x + 1)^2]/4 - [(y + 2)^2]/9 = 1B) [(x - 1)^2]/4 - [(y - 2)^2]/9 = 1C) [(x + 1)^2]/4 + [(y + 2)^2]/9 = 1D) [(x - 1)^2]/4 + [(y - 2)^2]/9 = 1My derived equation is [(x - 1)^2]/4 + [(y - 2)^2]/9 = 1, which is exactly option D. So, that must be the correct answer.Wait, let me double-check to make sure I didn't make any mistakes. I started by solving for cos θ and sin θ correctly, substituted them into the Pythagorean identity, and simplified. The signs in the equation are all correct because when I solved for cos θ and sin θ, I subtracted the constants from x and y, so the signs inside the squares are negative, which matches option D.Also, the denominators are 4 and 9, which correspond to the squares of the coefficients 2 and 3 from the original parametric equations. That makes sense because in the parametric equations, x is shifted by 1 and scaled by 2, and y is shifted by 2 and scaled by 3.Therefore, I'm confident that option D is the correct Cartesian equation for the given parametric equations.**Final Answer**boxed{D}

💡First, I need to determine the total cost price for the 300 metres of cloth. Given that the cost price per metre is Rs. 36, I can calculate the total cost price by multiplying the cost price per metre by the total number of metres.Next, I know the total selling price for the 300 metres of cloth is Rs. 9000. To find out the total loss, I subtract the total selling price from the total cost price.Finally, to find the loss per metre, I divide the total loss by the total number of metres sold. This will give me the loss incurred per metre of cloth.

💡First, I need to convert 50 kilograms into tons. Since 1 ton is equal to 1000 kilograms, I divide 50 by 1000 to get 0.05 tons.Next, I add the 0.05 tons to the 4 tons to find the total weight in tons. This gives me 4.05 tons.Finally, I compare the calculated value of 4.05 tons to the given value of 4.5 tons. Since they are not equal, the statement is false.

💡First, I need to express the number 5690 in scientific notation. Scientific notation requires the number to be written as a product of a number between 1 and 10 and a power of 10.To convert 5690 into this format, I will move the decimal point three places to the left. This gives me 5.690, which simplifies to 5.69.Next, I need to determine the appropriate power of 10. Since I moved the decimal point three places to the left, the power of 10 will be 10³.Combining these, the number 5690 in scientific notation is 5.69 × 10³.Looking at the options provided:- A: 56.9 × 10²- B: 5.69 × 10²- C: 5.69 × 10³- D: 0.569 × 10⁴The correct expression matches option C.

💡Okay, so I'm trying to figure out how many toucans are on the tree limb after one more joins the two that are already there. Let me break this down step by step to make sure I understand it correctly.First, there are 2 toucans sitting on the tree limb. That's straightforward. Now, the problem says that 1 more toucan joins them. So, I need to add that one toucan to the existing two. Let me visualize this. Imagine two toucans perched on a branch. They're probably making some noise or maybe just resting. Then, another toucan flies over and lands on the same branch. Now, how many toucans are there in total?To find the total number, I can use basic addition. I have 2 toucans initially, and then I add 1 more. So, 2 plus 1 equals... hmm, that should be 3. Wait, is there any chance I'm missing something here? Maybe the problem is trying to trick me somehow? Let me think. It says "1 more toucan joins them," which implies that the new toucan is joining the existing group. There's no mention of any toucans leaving or anything like that. So, it's just a simple addition problem.I guess I could also think about it in terms of objects. If I have two apples and someone gives me one more apple, how many apples do I have? It's the same principle. Two plus one equals three. Maybe I should write it out to be sure. Number of toucans initially = 2Number of toucans joining = 1Total number of toucans = 2 + 1 = 3Yeah, that seems right. I don't think there's any complexity here. It's a straightforward addition problem. I wonder if there's a way to represent this visually. Maybe drawing two toucans and then adding one more to see the total. That could help someone who's just learning to understand the concept better. Also, thinking about it in terms of real-life scenarios might make it clearer. If I have two friends at my house and one more friend comes over, how many friends are there in total? It's the same idea. I guess the key here is to recognize that when something "joins" a group, it means you're adding to the existing number. So, in this case, adding one to two gives you three. Is there any other way to approach this problem? Maybe using counting? Starting from one, two, and then adding another one makes it three. That works too. I think I've covered all the angles here. There's no trick or hidden detail in the problem. It's just a simple addition of one to two, resulting in three. So, to sum it up, there are three toucans on the tree limb after the additional one joins the original two.