answer：First, I need to understand the relationship between Alice's and Bob's initial amounts of money. It's given that Bob has 10% less money than Alice. This means if Alice has ( A ) dollars, Bob has ( 0.9A ) dollars.Next, after Alice buys an item, she has 10% less money than Bob. This implies that Alice's remaining money is 90% of Bob's money. So, Alice's remaining money can be expressed as ( 0.9 times 0.9A = 0.81A ).To find out how much money Alice spent, I subtract her remaining money from her initial amount: ( A - 0.81A = 0.19A ).Finally, to determine the percentage of her money that Alice spent, I calculate ( frac{0.19A}{A} times 100% = 19% ).

answer：Alright, so I have this problem about Li Ming's app, and I need to figure out how many days it takes for the user count to exceed 50,000. Let me try to break this down step by step.First, the problem gives me the user count function: A(t) = 500e^{kt}, where t is the number of days and k is a constant. It also tells me that after 10 days, there are 2000 users. My goal is to find the smallest t such that A(t) > 50,000.Okay, so let's start by using the information given for t = 10 days. At that point, the user count is 2000. So I can set up the equation:500e^{10k} = 2000I need to solve for k. Let me divide both sides by 500 to simplify:e^{10k} = 2000 / 500e^{10k} = 4Now, to solve for k, I can take the natural logarithm of both sides. Remember, ln(e^{x}) = x. So:ln(e^{10k}) = ln(4)10k = ln(4)Therefore, k = ln(4) / 10Hmm, I might need to calculate ln(4) or at least express it in terms of ln(2) since I know that ln(4) = 2ln(2). But wait, the problem gives me lg(2) = 0.30, which is the logarithm base 10, not the natural logarithm. Maybe I can relate them somehow.I recall that ln(x) = lg(x) / lg(e). So, ln(4) = lg(4) / lg(e). Let me compute lg(4) first. Since 4 is 2 squared, lg(4) = 2lg(2) = 2 * 0.30 = 0.60.So, ln(4) = 0.60 / lg(e). But I don't know lg(e). Wait, I know that ln(10) = 1 / lg(e) because ln(10) = 1 / (log_{10} e). So, lg(e) = 1 / ln(10). But I don't have ln(10) either. Maybe I can approximate it.I remember that ln(10) is approximately 2.3026, but since I don't have that value, maybe I can use the given lg(2) to approximate ln(2). Wait, let's see.Alternatively, maybe I can use the relation between ln and lg to express everything in terms of lg. Let me think.Given that A(t) = 500e^{kt}, and I need to find t when A(t) = 50,000.So, setting up the equation:500e^{kt} = 50,000Divide both sides by 500:e^{kt} = 100Take the natural logarithm of both sides:kt = ln(100)So, t = ln(100) / kBut I already have k = ln(4) / 10, so substituting:t = ln(100) / (ln(4) / 10) = (10 ln(100)) / ln(4)Hmm, that seems a bit complicated. Maybe I can express everything in terms of lg instead of natural logs.Let me try taking the logarithm base 10 of both sides of the original equation.Starting again, A(t) = 500e^{kt}At t = 10, A(10) = 2000:500e^{10k} = 2000Divide by 500:e^{10k} = 4Take log base 10:lg(e^{10k}) = lg(4)Using the logarithm power rule:10k * lg(e) = lg(4)We know lg(4) = 0.60, so:10k * lg(e) = 0.60Therefore, k = 0.60 / (10 * lg(e)) = 0.06 / lg(e)But I still don't know lg(e). Maybe I can find it using the relation between ln and lg.I know that ln(10) = 1 / lg(e), so lg(e) = 1 / ln(10). But without knowing ln(10), this might not help directly.Wait, maybe I can use the given lg(2) = 0.30 to approximate ln(2). Since ln(2) ≈ 0.6931, but since I don't have that, maybe I can relate it.Alternatively, perhaps I can use the change of base formula for logarithms.I know that ln(x) = lg(x) / lg(e). So, ln(4) = lg(4) / lg(e) = 0.60 / lg(e)Similarly, ln(100) = lg(100) / lg(e) = 2 / lg(e)So, going back to t = (10 ln(100)) / ln(4)Substituting the expressions in terms of lg:t = (10 * (2 / lg(e))) / (0.60 / lg(e)) ) = (20 / lg(e)) / (0.60 / lg(e)) ) = 20 / 0.60 = 33.333...So, t ≈ 33.333 days.Since the question asks for the number of days it takes to exceed 50,000, and t must be at least this value, we need to round up to the next whole number, which is 34 days.Wait, but let me double-check my steps to make sure I didn't make a mistake.1. Found k using t=10, A=2000: e^{10k}=4 => 10k=ln(4) => k=ln(4)/102. For A(t)=50,000: e^{kt}=100 => kt=ln(100) => t=ln(100)/k = ln(100)/(ln(4)/10)=10 ln(100)/ln(4)3. Expressed ln(100) and ln(4) in terms of lg: ln(100)=2/lg(e), ln(4)=0.60/lg(e)4. So t=10*(2/lg(e))/(0.60/lg(e))=10*2/0.60=20/0.60=33.333...5. Rounded up to 34 days.Yes, that seems correct. So the answer should be 34 days, which is option D.

answer：Okay, so I've got this problem here where a function ( f(x) ) satisfies the equation ( f(x - y) = f(x)g(y) ) for all real numbers ( x ) and ( y ). They also tell me that ( g(y) = e^{-y} ) and that ( f(x) ) is never zero for any real ( x ). I need to find ( f(5) ).Hmm, let me start by understanding what this equation is saying. It relates the value of the function at ( x - y ) to the value at ( x ) multiplied by ( g(y) ). Since ( g(y) ) is given as ( e^{-y} ), maybe I can substitute that in and see what happens.So, substituting ( g(y) ) into the equation, I get:[ f(x - y) = f(x) e^{-y} ]This has to hold true for all real numbers ( x ) and ( y ). That seems like a functional equation, which probably has some standard solutions. I remember that exponential functions often satisfy such multiplicative properties, so maybe ( f(x) ) is an exponential function.Let me think about what kind of function ( f(x) ) could be. If I consider ( f(x) ) as ( e^{kx} ) for some constant ( k ), then let's test if that works.Suppose ( f(x) = e^{kx} ). Then, substituting into the equation:[ f(x - y) = e^{k(x - y)} = e^{kx} e^{-ky} ]On the other hand, ( f(x)g(y) = e^{kx} e^{-y} ).So, for these two expressions to be equal for all ( x ) and ( y ), we must have:[ e^{kx} e^{-ky} = e^{kx} e^{-y} ]Dividing both sides by ( e^{kx} ) (which is never zero, so that's safe), we get:[ e^{-ky} = e^{-y} ]This implies that ( -ky = -y ) for all ( y ), so ( k = 1 ).Therefore, ( f(x) = e^{x} ) is a solution. Let me check if this satisfies the original equation:[ f(x - y) = e^{x - y} = e^{x} e^{-y} = f(x)g(y) ]Yes, that works. So, ( f(x) = e^{x} ) is indeed a solution.Wait, but the problem says ( f(x) neq 0 ) for all real ( x ). Well, ( e^{x} ) is always positive, so it's never zero. That fits the condition.So, if ( f(x) = e^{x} ), then ( f(5) = e^{5} ). But hold on, the assistant's initial thought process had ( f(5) = e^{2.5} ). That seems different. Did I make a mistake?Let me double-check. If I set ( x = 5 ) and ( y = 2.5 ), then:[ f(5 - 2.5) = f(5)g(2.5) ][ f(2.5) = f(5) e^{-2.5} ]So, ( f(2.5) = f(5) e^{-2.5} ). If I solve for ( f(5) ), I get:[ f(5) = f(2.5) e^{2.5} ]But then, if I substitute ( f(2.5) ) back into itself, it seems like I get an identity:[ f(2.5) = f(5) e^{-2.5} ][ f(2.5) = (f(2.5) e^{2.5}) e^{-2.5} ][ f(2.5) = f(2.5) ]Which is always true, but doesn't help me find a specific value for ( f(5) ). So, maybe there's a different approach.Wait, earlier I assumed ( f(x) = e^{x} ) and got ( f(5) = e^{5} ). But the assistant's thought process led to ( e^{2.5} ). Maybe I need to consider a different form for ( f(x) ).Let me think again. The functional equation is ( f(x - y) = f(x) e^{-y} ). Maybe I can fix ( x ) and vary ( y ), or fix ( y ) and vary ( x ). Let's see.If I fix ( x ) and let ( y ) vary, then for each fixed ( x ), ( f(x - y) ) is equal to ( f(x) e^{-y} ). This suggests that ( f ) is an exponential function because the right-hand side is exponential in ( y ).Alternatively, if I fix ( y ) and vary ( x ), then ( f(x - y) = f(x) e^{-y} ). This looks like a functional equation where the function at a shifted argument is proportional to the function at the original argument, scaled by an exponential factor. This is characteristic of exponential functions.So, perhaps ( f(x) ) is of the form ( f(x) = C e^{kx} ) for some constants ( C ) and ( k ). Let me test this.Let ( f(x) = C e^{kx} ). Then:[ f(x - y) = C e^{k(x - y)} = C e^{kx} e^{-ky} ]And:[ f(x) g(y) = C e^{kx} e^{-y} ]Setting these equal:[ C e^{kx} e^{-ky} = C e^{kx} e^{-y} ]Dividing both sides by ( C e^{kx} ) (which is non-zero):[ e^{-ky} = e^{-y} ]Which implies ( -ky = -y ) for all ( y ), so ( k = 1 ). Therefore, ( f(x) = C e^{x} ).Since ( f(x) ) is given to be non-zero for all ( x ), ( C ) must be non-zero. But the problem doesn't specify any particular value for ( f(x) ) other than it's non-zero, so ( C ) could be any non-zero constant. However, the problem asks for ( f(5) ). If ( f(x) = C e^{x} ), then ( f(5) = C e^{5} ).But wait, the problem doesn't give any initial condition like ( f(0) ) or anything else. So, without additional information, ( f(5) ) could be any multiple of ( e^{5} ). But the assistant's thought process concluded ( f(5) = e^{2.5} ). That suggests maybe I missed something.Let me go back to the original equation:[ f(x - y) = f(x) e^{-y} ]If I set ( x = y ), then:[ f(0) = f(x) e^{-x} ]So, ( f(x) = f(0) e^{x} ). That gives ( f(x) ) in terms of ( f(0) ). Therefore, ( f(5) = f(0) e^{5} ).But the problem doesn't give ( f(0) ). Unless ( f(0) ) can be determined from the given conditions. Let's see.From the equation ( f(x - y) = f(x) e^{-y} ), if I set ( y = 0 ), then:[ f(x - 0) = f(x) e^{0} ][ f(x) = f(x) cdot 1 ]Which is just an identity, doesn't give new information.If I set ( x = 0 ), then:[ f(0 - y) = f(0) e^{-y} ][ f(-y) = f(0) e^{-y} ]But since ( f(-y) = f(0) e^{-y} ), and earlier we had ( f(x) = f(0) e^{x} ), let's check consistency.If ( f(x) = f(0) e^{x} ), then ( f(-y) = f(0) e^{-y} ), which matches the equation above. So, that's consistent.Therefore, ( f(x) = f(0) e^{x} ), and ( f(5) = f(0) e^{5} ). But without knowing ( f(0) ), I can't determine the exact value of ( f(5) ). However, the problem states that ( f(x) neq 0 ) for all ( x ), which means ( f(0) neq 0 ). But it doesn't specify any particular value.Wait, maybe I can find ( f(0) ) by using the functional equation. Let me try setting ( x = y ), as I did before:[ f(0) = f(x) e^{-x} ]But this is true for all ( x ), so if I set ( x = 0 ), then:[ f(0) = f(0) e^{0} ][ f(0) = f(0) cdot 1 ]Which is just an identity again. So, no help.Alternatively, maybe I can differentiate both sides with respect to ( y ) or ( x ), but the problem doesn't state that ( f ) is differentiable. So, I shouldn't assume that.Wait, perhaps the problem expects a specific form without the constant ( C ). Maybe it's assuming ( f(0) = 1 ). If that's the case, then ( f(x) = e^{x} ), and ( f(5) = e^{5} ).But the assistant's thought process led to ( e^{2.5} ). Maybe there's a different approach. Let me see.In the assistant's thought process, they set ( x = 5 ) and ( y = 2.5 ), leading to:[ f(2.5) = f(5) e^{-2.5} ]Then, they substituted ( f(2.5) = f(5) e^{-2.5} ) back into itself, which just gave an identity. They then assumed ( f(5) = a e^{2.5} ), leading to ( f(5) = e^{2.5} ).But wait, if ( f(x) = e^{x} ), then ( f(5) = e^{5} ), and ( f(2.5) = e^{2.5} ). So, substituting back, ( e^{2.5} = e^{5} e^{-2.5} ), which simplifies to ( e^{2.5} = e^{2.5} ), which is true. So, the assistant's approach didn't actually determine ( f(5) ), it just showed that if ( f(5) = a e^{2.5} ), then ( a ) can be any constant, but without additional information, ( a ) remains arbitrary.Therefore, unless there's a specific condition given, ( f(5) ) could be any multiple of ( e^{5} ). But the problem states ( f(x) neq 0 ), so ( f(5) ) is non-zero, but its exact value isn't determined unless we assume ( f(0) = 1 ), which isn't given.Wait, maybe I'm overcomplicating. Let me think again. The functional equation is ( f(x - y) = f(x) e^{-y} ). If I set ( y = x ), then:[ f(0) = f(x) e^{-x} ]So, ( f(x) = f(0) e^{x} ). Therefore, ( f(5) = f(0) e^{5} ). But without knowing ( f(0) ), I can't find the exact value. However, maybe the problem expects the answer in terms of ( f(0) ), but it doesn't specify. Alternatively, perhaps ( f(0) = 1 ) is assumed, making ( f(5) = e^{5} ).But the assistant's thought process led to ( e^{2.5} ), which suggests maybe a different approach. Let me see.Alternatively, maybe the function is ( f(x) = e^{-x} ). Let's test that. Then:[ f(x - y) = e^{-(x - y)} = e^{-x} e^{y} ]And:[ f(x) g(y) = e^{-x} e^{-y} ]These are not equal unless ( e^{y} = e^{-y} ), which is only true for ( y = 0 ). So, that doesn't work.Alternatively, maybe ( f(x) = e^{kx} ) with ( k ) not necessarily 1. Let's see.Wait, earlier I concluded ( k = 1 ), so ( f(x) = C e^{x} ). Therefore, ( f(5) = C e^{5} ). But without knowing ( C ), I can't determine ( f(5) ).Wait, maybe the problem is designed so that ( f(5) ) is ( e^{5} ), assuming ( C = 1 ). But the assistant's thought process led to ( e^{2.5} ), which is ( sqrt{e^{5}} ). That seems inconsistent.Alternatively, maybe the function is ( f(x) = e^{-x} ), but as I saw earlier, that doesn't satisfy the equation unless ( y = 0 ).Wait, let me try substituting ( f(x) = e^{x} ) into the original equation again:[ f(x - y) = e^{x - y} = e^{x} e^{-y} = f(x) g(y) ]Yes, that works. So, ( f(x) = e^{x} ) is a solution, and thus ( f(5) = e^{5} ).But the assistant's thought process led to ( e^{2.5} ). Maybe they made a mistake. Let me see.In the assistant's thought process, they set ( x = 5 ) and ( y = 2.5 ), getting:[ f(2.5) = f(5) e^{-2.5} ]Then, they substituted ( f(2.5) = f(5) e^{-2.5} ) back into itself, which just gave an identity. They then assumed ( f(5) = a e^{2.5} ), leading to ( f(5) = e^{2.5} ).But this approach doesn't actually determine ( f(5) ), because ( a ) can be any constant. So, unless there's an additional condition, ( f(5) ) isn't uniquely determined. Therefore, the assistant's conclusion that ( f(5) = e^{2.5} ) might be incorrect because it assumes ( a = 1 ), which isn't justified by the problem statement.Therefore, the correct approach is to recognize that ( f(x) = C e^{x} ), and thus ( f(5) = C e^{5} ). However, without knowing ( C ), we can't determine ( f(5) ) exactly. But since the problem doesn't provide any additional conditions, perhaps it's expecting the answer in terms of ( f(0) ), or assuming ( f(0) = 1 ), which would make ( f(5) = e^{5} ).Alternatively, maybe the problem is designed to recognize that ( f(x) = e^{x} ) is the solution, hence ( f(5) = e^{5} ).Wait, but the assistant's thought process led to ( e^{2.5} ), which is ( sqrt{e^{5}} ). That seems like a different approach. Maybe they considered a different functional form.Alternatively, perhaps the function is ( f(x) = e^{-x} ), but as I saw earlier, that doesn't satisfy the equation unless ( y = 0 ).Wait, let me try a different approach. Let me assume that ( f(x) ) is of the form ( f(x) = e^{kx} ), then from the equation:[ e^{k(x - y)} = e^{kx} e^{-y} ]Which simplifies to:[ e^{kx} e^{-ky} = e^{kx} e^{-y} ]Dividing both sides by ( e^{kx} ):[ e^{-ky} = e^{-y} ]Thus, ( -ky = -y ) for all ( y ), so ( k = 1 ). Therefore, ( f(x) = C e^{x} ).So, ( f(5) = C e^{5} ). But without knowing ( C ), I can't find the exact value. However, the problem states ( f(x) neq 0 ), so ( C neq 0 ). But it doesn't specify ( C ).Wait, maybe the problem expects the answer in terms of ( f(0) ), which is ( C ). So, ( f(5) = f(0) e^{5} ). But since ( f(0) ) isn't given, perhaps the answer is expressed as ( e^{5} ) times some constant, but the problem asks for ( f(5) ), not in terms of ( f(0) ).Alternatively, maybe the problem assumes ( f(0) = 1 ), which would make ( f(5) = e^{5} ).But the assistant's thought process led to ( e^{2.5} ), which is different. I think the assistant might have made a mistake by assuming ( f(5) = a e^{2.5} ) without justification, leading to an incorrect conclusion.Therefore, the correct answer should be ( f(5) = e^{5} ), assuming ( f(0) = 1 ). But since the problem doesn't specify ( f(0) ), perhaps the answer is left in terms of ( f(0) ), but the problem asks for a numerical value, so likely ( e^{5} ).Wait, but the assistant's thought process led to ( e^{2.5} ), which is ( sqrt{e^{5}} ). That seems like a different approach, but I can't see how that would be justified without additional information.Alternatively, maybe the function is ( f(x) = e^{-x} ), but as I saw earlier, that doesn't satisfy the equation unless ( y = 0 ).Wait, let me try substituting ( f(x) = e^{-x} ) into the equation:[ f(x - y) = e^{-(x - y)} = e^{-x} e^{y} ]And:[ f(x) g(y) = e^{-x} e^{-y} ]These are equal only if ( e^{y} = e^{-y} ), which is only true for ( y = 0 ). So, that doesn't work.Therefore, the only solution is ( f(x) = C e^{x} ), leading to ( f(5) = C e^{5} ). Since ( C ) is non-zero, but not specified, the problem might expect the answer in terms of ( e^{5} ), assuming ( C = 1 ).Alternatively, maybe the problem is designed to recognize that ( f(x) = e^{x} ), hence ( f(5) = e^{5} ).But the assistant's thought process led to ( e^{2.5} ), which is inconsistent with this. Therefore, I think the correct answer is ( f(5) = e^{5} ).

answer：To find the remainder when the sum (90 + 91 + 92 + 93 + 94 + 95 + 96 + 97) is divided by 17, I can start by calculating the sum of these numbers.First, I'll add the numbers sequentially:- (90 + 91 = 181)- (181 + 92 = 273)- (273 + 93 = 366)- (366 + 94 = 460)- (460 + 95 = 555)- (555 + 96 = 651)- (651 + 97 = 748)So, the total sum is 748.Next, I need to determine the remainder when 748 is divided by 17. I'll perform the division:- (748 ÷ 17 = 44) with a remainder of 0.Therefore, the remainder when the sum is divided by 17 is 0.

answer：Okay, so I have this problem where there are four points: P, Q, R, and S in space. The lines SP, SQ, and SR are each perpendicular to the other two. That means SP is perpendicular to SQ, SP is perpendicular to SR, and SQ is perpendicular to SR. Interesting. So, S is like the origin point, and P, Q, R are each along different axes from S, right?They give me the lengths: SP and SQ are both 12 units, and SR is 11 units. I need to find the volume of the pyramid SPQR. Hmm, okay. So, pyramids have a base and a height, and the volume is (1/3)*base area*height. But I need to figure out what the base is and what the height is in this case.Since SP, SQ, and SR are all perpendicular, it reminds me of a 3D coordinate system where S is the origin, and P, Q, R are along the x, y, z axes respectively. So, if I imagine S at (0,0,0), then P could be at (12,0,0), Q at (0,12,0), and R at (0,0,11). Is that right? Yeah, because each of those lines is perpendicular to the others.So, the pyramid SPQR would have its base as triangle PQR and the apex at S. Wait, or is the base another face? Hmm, actually, in a pyramid, the base can be any face, but usually, it's the one that's a polygon, and the apex is the point not on the base. So, in this case, since S is connected to P, Q, R, the base is triangle PQR, and S is the apex. So, the volume would be (1/3)*area of base*height.But wait, I need to figure out the area of the base, which is triangle PQR, and then the height from S to that base. Hmm, but since SP, SQ, SR are all perpendicular, maybe there's a simpler way to calculate the volume without having to find the area of triangle PQR and the height separately.I remember that for a rectangular prism, the volume is length*width*height, but this isn't a prism; it's a pyramid. However, since SP, SQ, and SR are mutually perpendicular, maybe the volume can be calculated using those lengths. Let me think.In a way, the pyramid is formed by three edges meeting at S, each perpendicular to the other two. So, the volume formula for such a pyramid is (1/6)*SP*SQ*SR. Is that correct? Wait, let me verify.If I consider the rectangular prism formed by SP, SQ, and SR, its volume would be SP*SQ*SR, which is 12*12*11. But the pyramid is just a fraction of that prism. How much exactly? I think it's 1/6 of the prism because the pyramid occupies one of the six possible pyramids that make up the prism when you divide it along the space diagonals.So, if the prism's volume is 12*12*11 = 1584, then the pyramid's volume would be 1584/6 = 264. That seems right. But let me make sure.Alternatively, I can think of the pyramid as having a base that's a right triangle in 3D space. The base is triangle PQR, and the apex is S. To find the volume, I need the area of the base and the height from S to the base.First, let's find the coordinates of P, Q, R, and S to better visualize this. Let me assign coordinates:- Let S be at (0,0,0).- Since SP is 12 and along the x-axis, P is at (12,0,0).- SQ is 12 and along the y-axis, so Q is at (0,12,0).- SR is 11 and along the z-axis, so R is at (0,0,11).So, the points are:- S: (0,0,0)- P: (12,0,0)- Q: (0,12,0)- R: (0,0,11)Now, the base is triangle PQR. Let's find the coordinates of P, Q, R:- P: (12,0,0)- Q: (0,12,0)- R: (0,0,11)To find the area of triangle PQR, I can use the formula for the area of a triangle given by three points in 3D space. The formula is (1/2)*|cross product of two sides|.First, let's find two vectors in the plane of PQR. Let's take vectors PQ and PR.Vector PQ = Q - P = (0-12, 12-0, 0-0) = (-12,12,0)Vector PR = R - P = (0-12, 0-0, 11-0) = (-12,0,11)Now, compute the cross product of PQ and PR.Cross product PQ × PR is:|i   j   k||-12 12   0||-12 0   11|= i*(12*11 - 0*0) - j*(-12*11 - 0*(-12)) + k*(-12*0 - 12*(-12))= i*(132) - j*(-132) + k*(144)= (132, 132, 144)Now, the magnitude of this cross product is sqrt(132^2 + 132^2 + 144^2).Calculating each term:132^2 = 17,424132^2 = 17,424144^2 = 20,736Adding them up: 17,424 + 17,424 + 20,736 = 55,584So, the magnitude is sqrt(55,584). Let me compute that.sqrt(55,584) = 235.7 approximately, but let me see if it's a perfect square.Wait, 235^2 = 55,225 and 236^2 = 55,696. So, 55,584 is between these. Hmm, maybe I made a mistake in the cross product.Wait, let me recalculate the cross product.PQ = (-12,12,0)PR = (-12,0,11)Cross product:i*(12*11 - 0*0) = i*(132 - 0) = 132ij*( - ( (-12)*11 - 0*(-12) )) = j*( - ( -132 - 0 )) = j*(132)k*( (-12)*0 - 12*(-12) ) = k*(0 + 144) = 144kSo, cross product is (132, 132, 144). So, the magnitude is sqrt(132^2 + 132^2 + 144^2) = sqrt(17,424 + 17,424 + 20,736) = sqrt(55,584). Hmm, 55,584 divided by 144 is 386.5, which is not a perfect square. Maybe I can factor it.Wait, 55,584 = 16 * 3,474. Hmm, 3,474 divided by 9 is 386. So, 55,584 = 16 * 9 * 386 = 144 * 386. So, sqrt(144 * 386) = 12*sqrt(386). Hmm, 386 is 2*193, which is prime. So, sqrt(55,584) = 12*sqrt(386). That's approximately 12*19.646 = 235.75.So, the area of triangle PQR is (1/2)*235.75 ≈ 117.875. Hmm, but that seems complicated. Maybe there's a better way.Wait, maybe I should use the formula for the volume of a tetrahedron. Since S, P, Q, R are points in space with SP, SQ, SR mutually perpendicular, the volume can be calculated as (1/6)*|SP · (SQ × SR)|. Since SP, SQ, SR are mutually perpendicular, the scalar triple product is just SP*SQ*SR.So, volume = (1/6)*12*12*11 = (1/6)*1584 = 264. That's the same answer as before. So, maybe I didn't need to go through all that cross product business.But just to make sure, let's think about it again. The scalar triple product gives the volume of the parallelepiped formed by the vectors SP, SQ, SR. Since the pyramid is 1/6 of that parallelepiped, the volume is (1/6)*12*12*11 = 264. Yep, that makes sense.Alternatively, if I consider the base as the triangle SPQ, which is a right triangle with legs 12 and 12, its area is (1/2)*12*12 = 72. Then, the height from R to this base would be along SR, which is 11. But wait, is that correct? Because in 3D, the height isn't just SR; it's the perpendicular distance from R to the plane SPQ.Wait, actually, since SP, SQ, SR are mutually perpendicular, the height from R to the base SPQ is indeed SR, because SR is perpendicular to both SP and SQ, which lie on the base. So, the height is 11. Therefore, the volume is (1/3)*72*11 = (1/3)*792 = 264. Yep, same answer.So, whether I use the scalar triple product method or the base area and height method, I get the same result. Therefore, the volume of pyramid SPQR is 264 cubic units.

answer：First, I need to find the percentage of (x + y) given that 20% of (x - y) equals this unknown percentage of (x + y). I'll denote this unknown percentage as P%.I know that y is 17.647058823529413% of x, which I can express as y = 0.17647058823529413 * x.Next, I'll substitute this expression for y into the original equation:0.20 * (x - y) = (P/100) * (x + y)Substituting y:0.20 * (x - 0.17647058823529413 * x) = (P/100) * (x + 0.17647058823529413 * x)Simplifying the equation:0.20 * (1 - 0.17647058823529413) * x = (P/100) * (1 + 0.17647058823529413) * xCalculating the values:0.20 * 0.8235294117647059 * x = (P/100) * 1.1764705882352942 * xThis simplifies to:0.16470588235294118 * x = (P/100) * 1.1764705882352942 * xDividing both sides by x:0.16470588235294118 = (P/100) * 1.1764705882352942Finally, solving for P:P = (0.16470588235294118 / 1.1764705882352942) * 100 ≈ 14%So, the percentage of (x + y) is approximately 14%.