answer：First, I need to determine the original cost price of the car. Let's denote this original cost price as ( x ).The man sells the car to his friend at a 15% loss. This means the selling price is 85% of the original cost price. Therefore, the selling price to the friend is:[text{Selling Price} = 0.85 times x]The friend then sells the car for Rs. 54,000, making a 20% gain. This means the selling price is 120% of the price the friend bought it for. Let's denote the price the friend bought it for as ( y ). So, we have:[1.20 times y = 54,000]Since ( y ) is equal to the selling price from the man, which is ( 0.85 times x ), we can substitute ( y ) with ( 0.85 times x ):[1.20 times (0.85 times x) = 54,000]Now, let's solve for ( x ):[1.20 times 0.85 times x = 54,000][1.02 times x = 54,000][x = frac{54,000}{1.02}][x = 52,941.18]Therefore, the original cost price of the car was approximately Rs. 52,941.18.

answer：Okay, so I have this problem where there's a cube with edge length 1, and inside it, there's an inscribed sphere called O₁. Then, there's another smaller sphere, O₂, inside the cube. This smaller sphere is externally tangent to the larger sphere O₁ and also tangent to three faces of the cube. I need to find the surface area of this smaller sphere O₂.First, let me visualize the cube and the spheres. The cube has edge length 1, so each face is a square with sides of length 1. The inscribed sphere O₁ touches all the faces of the cube. Since it's inscribed, its diameter should be equal to the edge length of the cube. So, the radius of O₁ should be half of that, which is 1/2.Now, the smaller sphere O₂ is inside the cube too. It's externally tangent to O₁, meaning they touch each other at exactly one point, and O₂ doesn't overlap with O₁. Additionally, O₂ is tangent to three faces of the cube. Since the cube has six faces, being tangent to three faces probably means it's touching one face on each of three adjacent sides, like the bottom, front, and left faces, for example.I think the key here is to figure out the radius of O₂. Once I have that, I can easily compute the surface area because the surface area of a sphere is 4πr².So, let's denote the radius of O₁ as r₁ and the radius of O₂ as r₂. As I thought earlier, r₁ is 1/2.Now, since O₂ is tangent to three faces of the cube, its center must be at a distance of r₂ from each of those three faces. If I consider the cube with coordinates from (0,0,0) to (1,1,1), then the center of O₂ would be at (r₂, r₂, r₂). Similarly, the center of O₁, being the inscribed sphere, is at the center of the cube, which is (0.5, 0.5, 0.5).The distance between the centers of O₁ and O₂ should be equal to the sum of their radii because they are externally tangent. So, the distance between (0.5, 0.5, 0.5) and (r₂, r₂, r₂) should be r₁ + r₂.Let me write that down:Distance between centers = sqrt[(0.5 - r₂)² + (0.5 - r₂)² + (0.5 - r₂)²] = r₁ + r₂Simplify the left side:sqrt[3*(0.5 - r₂)²] = r₁ + r₂Take the square root of 3*(0.5 - r₂)²:sqrt(3)*(0.5 - r₂) = r₁ + r₂We know r₁ is 0.5, so substitute that in:sqrt(3)*(0.5 - r₂) = 0.5 + r₂Now, let's solve for r₂.First, expand the left side:sqrt(3)*0.5 - sqrt(3)*r₂ = 0.5 + r₂Bring all terms involving r₂ to one side and constants to the other:sqrt(3)*0.5 - 0.5 = sqrt(3)*r₂ + r₂Factor out r₂ on the right side:sqrt(3)*0.5 - 0.5 = r₂*(sqrt(3) + 1)Now, factor out 0.5 on the left side:0.5*(sqrt(3) - 1) = r₂*(sqrt(3) + 1)Now, solve for r₂:r₂ = [0.5*(sqrt(3) - 1)] / (sqrt(3) + 1)To simplify this, let's rationalize the denominator. Multiply numerator and denominator by (sqrt(3) - 1):r₂ = [0.5*(sqrt(3) - 1)*(sqrt(3) - 1)] / [(sqrt(3) + 1)*(sqrt(3) - 1)]Simplify the denominator:(sqrt(3) + 1)*(sqrt(3) - 1) = (sqrt(3))² - (1)² = 3 - 1 = 2Now, the numerator:(sqrt(3) - 1)*(sqrt(3) - 1) = (sqrt(3))² - 2*sqrt(3) + 1 = 3 - 2*sqrt(3) + 1 = 4 - 2*sqrt(3)So, numerator becomes:0.5*(4 - 2*sqrt(3)) = 0.5*2*(2 - sqrt(3)) = (2 - sqrt(3))Therefore, r₂ = (2 - sqrt(3)) / 2So, the radius of the smaller sphere O₂ is (2 - sqrt(3))/2.Now, to find the surface area, use the formula 4πr²:Surface area = 4π[(2 - sqrt(3))/2]^2Let's compute that:First, square the radius:[(2 - sqrt(3))/2]^2 = (2 - sqrt(3))² / 4 = [4 - 4*sqrt(3) + 3] / 4 = (7 - 4*sqrt(3))/4Now, multiply by 4π:Surface area = 4π * (7 - 4*sqrt(3))/4 = π*(7 - 4*sqrt(3))So, the surface area of the smaller sphere O₂ is (7 - 4*sqrt(3))π.Wait, let me double-check my steps to make sure I didn't make any mistakes.1. Calculated r₁ correctly as 0.5.2. Set up the distance between centers correctly as sqrt[3*(0.5 - r₂)^2].3. Set that equal to r₁ + r₂, which makes sense because they are externally tangent.4. Expanded and solved for r₂, getting r₂ = (2 - sqrt(3))/2.5. Then, squared that and multiplied by 4π, simplifying to (7 - 4*sqrt(3))π.Everything seems to check out. So, I think that's the correct answer.**Final Answer**The surface area of the smaller sphere ( O_2 ) is boxed{(7 - 4sqrt{3})pi}.

answer：Okay, so I have this equation here: (10 - x)^2 equals x squared plus 6. Hmm, I need to find the value of x that makes this equation true. Alright, let's break this down step by step. First, I remember that when you have something squared like (10 - x)^2, you can expand it using the formula (a - b)^2 = a^2 - 2ab + b^2. So, applying that here, (10 - x)^2 should expand to 10 squared minus 2 times 10 times x plus x squared. Let me write that out:(10 - x)^2 = 10^2 - 2*10*x + x^2 = 100 - 20x + x^2.Okay, so the left side of the equation becomes 100 - 20x + x^2. The right side of the equation is already given as x^2 + 6. So now, my equation looks like this:100 - 20x + x^2 = x^2 + 6.Hmm, I notice that both sides have an x squared term. Maybe I can subtract x^2 from both sides to simplify the equation. Let's try that:100 - 20x + x^2 - x^2 = x^2 + 6 - x^2.Simplifying both sides, the x^2 terms cancel out:100 - 20x = 6.Alright, now I have a simpler equation: 100 minus 20x equals 6. I need to solve for x. Let me subtract 100 from both sides to isolate the term with x:100 - 20x - 100 = 6 - 100.This simplifies to:-20x = -94.Now, to solve for x, I should divide both sides by -20:x = (-94)/(-20).Dividing two negative numbers gives a positive result, so:x = 94/20.I can simplify this fraction by dividing both numerator and denominator by 2:94 ÷ 2 = 47,20 ÷ 2 = 10.So, x = 47/10, which is equal to 4.7.Wait, let me double-check my steps to make sure I didn't make a mistake. I expanded (10 - x)^2 correctly, subtracted x^2 from both sides, then subtracted 100, and finally divided by -20. Everything seems to add up. Just to be thorough, I'll plug x = 4.7 back into the original equation to verify:Left side: (10 - 4.7)^2 = (5.3)^2 = 28.09.Right side: (4.7)^2 + 6 = 22.09 + 6 = 28.09.Both sides are equal, so x = 4.7 is indeed the correct solution. I think I've got it!

answer：First, I need to determine the distance Nina ran twice during her drill. Let's denote this unknown distance as ( x ).According to the problem, Nina ran this distance twice and then ran an additional ( 0.6666666666666666 ) miles. The total distance she ran is ( 0.8333333333333334 ) miles. This can be represented by the equation:[2x + 0.6666666666666666 = 0.8333333333333334]To find ( x ), I'll first subtract ( 0.6666666666666666 ) from both sides of the equation:[2x = 0.8333333333333334 - 0.6666666666666666][2x = 0.16666666666666674]Next, I'll divide both sides by 2 to solve for ( x ):[x = frac{0.16666666666666674}{2}][x = 0.08333333333333337]Therefore, the distance Nina ran twice is ( 0.08333333333333337 ) miles.

answer：Okay, so I have this problem here about Mitzi going to an amusement park and spending some money. I need to figure out how much money she brought with her initially. Let me try to break this down step by step.First, the problem says she spent 30 on a ticket. That seems straightforward. Then, she spent 13 on food. Okay, so that's another expense. After that, she bought a T-shirt for 23. So far, she's spent 30 + 13 + 23. Hmm, let me add those up. 30 plus 13 is 43, and then 43 plus 23 is 66. So she's spent a total of 66.Now, the problem also mentions that she has 9 left after all these expenses. So, if she has 9 left and she spent 66, how much did she bring initially? I think I need to add the amount she spent to the amount she has left to find out the total amount she brought.So, 66 spent plus 9 left equals... let me do that addition. 66 plus 9 is 75. So, Mitzi must have brought 75 to the amusement park.Wait, let me double-check that to make sure I didn't make any mistakes. She spent 30 on a ticket, 13 on food, and 23 on a T-shirt. Adding those together: 30 + 13 is 43, and 43 + 23 is indeed 66. She has 9 left, so 66 + 9 equals 75. Yeah, that seems right.I guess another way to think about it is to consider her total money as the sum of her expenses and her remaining money. So, total money = expenses + remaining money. Plugging in the numbers: total money = 66 + 9 = 75. Yep, that makes sense.I don't think I missed anything here. The problem gives clear amounts for each expense and the remaining money, so it's just a matter of adding them up correctly. I feel confident that 75 is the right answer.

answer：Alright, so I have this problem here: "What is the maximum possible number of planes that can be determined by 9 points in space, assuming no four points are coplanar and no three are collinear?" Hmm, okay, let me try to figure this out step by step.First, I know that in geometry, a plane is determined by three non-collinear points. So, if I have three points that aren't all on the same straight line, they define a unique plane. Now, the problem says there are 9 points in space, and no three are collinear. That means every set of three points will form a plane, right? So, at first glance, it seems like I just need to calculate how many different sets of three points I can get from nine points.But wait, there's another condition: no four points are coplanar. That means that no four points lie on the same plane. So, if I just calculate the number of ways to choose three points from nine, that would give me the total number of planes, but I have to make sure that no four points are on the same plane. Hmm, does that affect the number of planes? Let me think.If no four points are coplanar, that means every plane can contain at most three of these points. So, each plane is uniquely determined by exactly three points, and there are no additional points lying on the same plane. That means I don't have to worry about overcounting planes because of multiple points lying on the same plane.Okay, so if I have 9 points, and I want to find out how many unique planes they determine, with the conditions that no three are collinear and no four are coplanar, then it's just the number of combinations of three points from nine. That should give me the total number of planes.The formula for combinations is C(n, k) = n! / (k!(n - k)!), where n is the total number of items, and k is the number of items to choose. So, in this case, n is 9 and k is 3.Let me calculate that:C(9, 3) = 9! / (3!(9 - 3)!) = 9! / (3!6!) = (9 × 8 × 7 × 6!) / (3 × 2 × 1 × 6!) = (9 × 8 × 7) / (3 × 2 × 1) = 504 / 6 = 84.So, that gives me 84. But wait, is that the maximum number of planes? Let me double-check.Since no four points are coplanar, each plane is determined by exactly three points, and there are no overlapping planes with more than three points. So, each combination of three points gives a unique plane, and there are no extra planes because of multiple points lying on the same plane. Therefore, 84 should indeed be the maximum number of planes.But just to be thorough, let me consider if there's any other way this could be approached. Maybe using some geometric intuition or another combinatorial method. Hmm, I don't think so. Since each plane is uniquely determined by three points, and there are no four points on the same plane, the number of planes is exactly the number of ways to choose three points from nine.Another way to think about it is, for each point, how many planes does it lie on? Well, each point can be combined with any two of the other eight points to form a plane. So, for one point, the number of planes it lies on is C(8, 2) = 28. Since there are nine points, that would give 9 × 28 = 252. But wait, that counts each plane three times because each plane is determined by three points, and each plane is being counted once for each of its three points. So, to get the actual number of unique planes, I need to divide by 3. So, 252 / 3 = 84. Yep, that matches my earlier calculation.Okay, so that seems consistent. Both methods give me 84. So, I think that's the right answer. The maximum number of planes determined by 9 points in space, with no four coplanar and no three collinear, is 84.Just to recap:1. Each plane is determined by exactly three non-collinear points.2. No four points lie on the same plane, so each plane is unique to its three points.3. The number of ways to choose three points from nine is C(9, 3) = 84.4. Alternatively, considering each point lies on C(8, 2) = 28 planes, and since each plane is counted three times, the total number of unique planes is (9 × 28) / 3 = 84.Therefore, I'm confident that the maximum number of planes is 84.