💡First, I recognize that the ratio of the edges of the two cubes is 3:1. This means that if the edge of the first cube is 3 units, the edge of the second cube is 1 unit.Next, I recall that the volume of a cube is calculated by cubing the length of its edge. So, the volume of the first cube is (3^3 = 27) units, which matches the given information.To find the volume of the second cube, I cube the edge length of the second cube: (1^3 = 1) unit.Therefore, the volume of the second cube is 1 unit.

💡Okay, so I have this problem here where I need to find the value of 2a - b. The functions given are f(x) = ax + b and g(x) = -5x + 7. Then, h(x) is defined as f composed with g, which is f(g(x)). Also, the inverse of h(x) is given as h^{-1}(x) = x - 9. Hmm, that seems a bit tricky, but let me try to break it down step by step.First, I know that h(x) is f(g(x)). So, I should substitute g(x) into f(x). Let me write that out:h(x) = f(g(x)) = f(-5x + 7)Since f(x) is ax + b, substituting -5x + 7 into f would give:h(x) = a*(-5x + 7) + bLet me compute that:h(x) = -5a*x + 7a + bSo, h(x) is a linear function with slope -5a and y-intercept (7a + b). Now, the inverse function h^{-1}(x) is given as x - 9. I remember that if h^{-1}(x) is the inverse of h(x), then applying h to h^{-1}(x) should give me x. In other words:h(h^{-1}(x)) = xLet me substitute h^{-1}(x) into h(x):h(h^{-1}(x)) = h(x - 9)Since h(x) is -5a*x + 7a + b, substituting x - 9 into h gives:h(x - 9) = -5a*(x - 9) + 7a + bLet me expand this:= -5a*x + 45a + 7a + bCombine like terms:= -5a*x + (45a + 7a) + b= -5a*x + 52a + bBut we know that h(h^{-1}(x)) should equal x, so:-5a*x + 52a + b = xNow, this equation must hold for all x, which means the coefficients of x on both sides must be equal, and the constant terms must also be equal. Let me write that out:For the coefficients of x:-5a = 1For the constant terms:52a + b = 0Okay, so I can solve these two equations to find a and b.Starting with the first equation:-5a = 1Divide both sides by -5:a = 1 / (-5)a = -1/5Now, plug this value of a into the second equation:52a + b = 052*(-1/5) + b = 0Compute 52*(-1/5):52/5 = 10.4, so 52*(-1/5) = -10.4So, -10.4 + b = 0Add 10.4 to both sides:b = 10.4But 10.4 is a decimal. Since the problem might prefer fractions, let me convert that. 10.4 is the same as 52/5 because 52 divided by 5 is 10.4.So, b = 52/5Now, I need to find 2a - b.First, compute 2a:2a = 2*(-1/5) = -2/5Then, subtract b:2a - b = (-2/5) - (52/5)Since both have the same denominator, I can combine them:= (-2 - 52)/5= (-54)/5So, 2a - b equals -54/5.Let me double-check my steps to make sure I didn't make a mistake.1. I found h(x) by substituting g(x) into f(x), which gave me -5a*x + 7a + b. That seems correct.2. Then, I used the fact that h(h^{-1}(x)) = x, so I substituted h^{-1}(x) into h(x). That gave me -5a*(x - 9) + 7a + b. Expanding that, I got -5a*x + 45a + 7a + b, which simplifies to -5a*x + 52a + b. That seems right.3. Setting that equal to x, I equated the coefficients and constants, leading to -5a = 1 and 52a + b = 0. Solving these, I got a = -1/5 and b = 52/5. That makes sense.4. Finally, computing 2a - b, I substituted the values and got -54/5. That seems consistent.I think I did everything correctly. So, the value of 2a - b is -54/5.

💡Alright, so I'm trying to figure out how much money Bözi and Rózsi had and what the approximate price of the scarves they bought was. Let's see, from their conversation, I can gather a few key points.First, Rózsi mentions that she needs 5 tizes (which is 50 forints) to buy a shawl, and even if she had 40 forints more, she still wouldn't have enough. So, if I let Rózsi's money be ( y ) forints, the price of the shawl ( s ) must satisfy:[ y + 40 < s < y + 50 ]Then, Bözi says that if she had 40 forints, she could buy the shawl and still have some money left for the cinema. Let's say Bözi's money is ( x ) forints, and the money left for the cinema is ( m ) forints, which is less than 10 forints. So, this gives us:[ x + 30 < s leq x + 40 - m ]Next, they talk about a second shawl that's 20% cheaper. The price of this shawl would be ( 0.8s ). Bözi mentions that with her money, she could buy this cheaper shawl and still have some left for the cinema. So:[ 0.8s leq x + 20 ]Rózsi also mentions that with her money, she could buy the cheaper shawl and have some left for the cinema:[ 0.8s leq y + 30 ]Then, they discuss a third shawl that's even cheaper, specifically ( 0.6s - 3 ) forints. Rózsi says she doesn't have enough for this one, but Bözi says she could buy it and still have some money left for the cinema. So:[ y < 0.6s - 3 < y + 10 ][ x - 10 < 0.6s - 3 < x ]Finally, they talk about how much money they have left after buying the shawls, which is ( v ) forints. So:[ x + y - 1.2s = v ]Now, I need to solve these inequalities to find the range of ( x ), ( y ), and ( s ).Starting with Rózsi's initial statement:[ y + 40 < s < y + 50 ]This tells me that the shawl costs more than Rózsi's money plus 40 forints but less than her money plus 50 forints.From Bözi's statement:[ x + 30 < s leq x + 40 - m ]Since ( m ) is less than 10, this means:[ x + 30 < s < x + 40 ]Combining these two, we have:[ y + 40 < s < x + 40 ]Which implies:[ y + 40 < x + 40 ]So:[ y < x ]This means Bözi has more money than Rózsi.Next, looking at the cheaper shawl:[ 0.8s leq x + 20 ][ 0.8s leq y + 30 ]From Rózsi's perspective:[ 0.8s - 4 = y + 20 ]So:[ y = 0.8s - 24 ]From Bözi's perspective:[ 0.8s leq x + 20 ]So:[ x geq 0.8s - 20 ]Now, for the third shawl:[ y < 0.6s - 3 < y + 10 ][ x - 10 < 0.6s - 3 < x ]Substituting ( y = 0.8s - 24 ) into the first inequality:[ 0.8s - 24 < 0.6s - 3 < 0.8s - 24 + 10 ]Simplifying:[ 0.8s - 24 < 0.6s - 3 ][ 0.2s < 21 ][ s < 105 ]And:[ 0.6s - 3 < 0.8s - 14 ][ -0.2s < -11 ][ s > 55 ]So, combining these, we have:[ 55 < s < 105 ]From Bözi's perspective on the third shawl:[ x - 10 < 0.6s - 3 < x ][ x - 10 < 0.6s - 3 ][ x < 0.6s + 7 ]And:[ 0.6s - 3 < x ]So:[ x > 0.6s - 3 ]Combining with the earlier inequality ( x geq 0.8s - 20 ), we have:[ 0.8s - 20 leq x < 0.6s + 7 ]This gives us:[ 0.8s - 20 < 0.6s + 7 ][ 0.2s < 27 ][ s < 135 ]But we already have ( s < 105 ), so this doesn't change our range.Now, considering the final statement about the remaining money:[ x + y - 1.2s = v ]Substituting ( y = 0.8s - 24 ):[ x + 0.8s - 24 - 1.2s = v ][ x - 0.4s - 24 = v ][ x = 0.4s + v + 24 ]Since ( v ) is the remaining money, it should be positive but not specified exactly. However, we can assume it's a reasonable amount, say between 1 and 10 forints.Let's assume ( v = 5 ) forints (just a reasonable guess):[ x = 0.4s + 5 + 24 ][ x = 0.4s + 29 ]Now, we have:[ x = 0.4s + 29 ]And from earlier:[ x > 0.6s - 3 ][ 0.4s + 29 > 0.6s - 3 ][ 0.2s < 32 ][ s < 160 ]Which is already satisfied since ( s < 105 ).Also:[ x < 0.6s + 7 ][ 0.4s + 29 < 0.6s + 7 ][ 0.2s > 22 ][ s > 110 ]But we have ( s < 105 ), which contradicts ( s > 110 ). This means our assumption of ( v = 5 ) might be incorrect.Let's try ( v = 10 ):[ x = 0.4s + 10 + 24 ][ x = 0.4s + 34 ]Now:[ x > 0.6s - 3 ][ 0.4s + 34 > 0.6s - 3 ][ 0.2s < 37 ][ s < 185 ]Still satisfied.And:[ x < 0.6s + 7 ][ 0.4s + 34 < 0.6s + 7 ][ 0.2s > 27 ][ s > 135 ]Again, contradicts ( s < 105 ).Hmm, seems like increasing ( v ) isn't helping. Maybe ( v ) is less than 5.Let's try ( v = 1 ):[ x = 0.4s + 1 + 24 ][ x = 0.4s + 25 ]Now:[ x > 0.6s - 3 ][ 0.4s + 25 > 0.6s - 3 ][ 0.2s < 28 ][ s < 140 ]Still okay.And:[ x < 0.6s + 7 ][ 0.4s + 25 < 0.6s + 7 ][ 0.2s > 18 ][ s > 90 ]So, combining with our earlier range:[ 90 < s < 105 ]This seems plausible.So, ( s ) is between 90 and 105 forints.Now, let's find ( x ) and ( y ).From ( y = 0.8s - 24 ):If ( s = 90 ):[ y = 0.8*90 - 24 = 72 - 24 = 48 ]If ( s = 105 ):[ y = 0.8*105 - 24 = 84 - 24 = 60 ]So, ( y ) is between 48 and 60 forints.From ( x = 0.4s + 25 ):If ( s = 90 ):[ x = 0.4*90 + 25 = 36 + 25 = 61 ]If ( s = 105 ):[ x = 0.4*105 + 25 = 42 + 25 = 67 ]So, ( x ) is between 61 and 67 forints.Therefore, Bözi had approximately 61 to 67 forints, and Rózsi had approximately 48 to 60 forints. The scarves they bought were priced between 90 and 105 forints.

💡Okay, so I have this problem where I need to find the product of all possible values of ( m ) given that ( p ), ( q ), and ( r ) are distinct complex numbers satisfying the equations:[frac{p}{1 - q} = frac{q}{1 - r} = frac{r}{1 - p} = m]Hmm, let me try to unpack this step by step. First, since all three fractions are equal to ( m ), I can express each variable in terms of the next one. That is:1. ( p = m(1 - q) )2. ( q = m(1 - r) )3. ( r = m(1 - p) )So, each variable is expressed in terms of another. This seems like a system of equations that I can solve by substitution. Let me try substituting one into another.Starting with the first equation, ( p = m(1 - q) ). But from the second equation, ( q = m(1 - r) ). So, substituting ( q ) into the first equation gives:[p = m(1 - m(1 - r))]Simplify that:[p = m - m^2(1 - r)][p = m - m^2 + m^2 r]Now, from the third equation, ( r = m(1 - p) ). Let's substitute this into the equation above:[p = m - m^2 + m^2 [m(1 - p)]][p = m - m^2 + m^3(1 - p)]Let me distribute the ( m^3 ):[p = m - m^2 + m^3 - m^3 p]Now, let's collect like terms. Bring all terms involving ( p ) to the left side:[p + m^3 p = m - m^2 + m^3]Factor out ( p ):[p(1 + m^3) = m - m^2 + m^3]Hmm, so:[p = frac{m - m^2 + m^3}{1 + m^3}]Wait, let me factor numerator and denominator if possible.The numerator is ( m^3 - m^2 + m ). Let me factor out an ( m ):[m(m^2 - m + 1)]The denominator is ( 1 + m^3 ). I remember that ( a^3 + b^3 = (a + b)(a^2 - ab + b^2) ). So, ( 1 + m^3 = (1 + m)(1 - m + m^2) ).So, substituting back:[p = frac{m(m^2 - m + 1)}{(1 + m)(1 - m + m^2)}]Wait, the numerator has ( m^2 - m + 1 ) and the denominator has ( 1 - m + m^2 ), which is the same as ( m^2 - m + 1 ). So, they cancel out:[p = frac{m}{1 + m}]Interesting! So, ( p = frac{m}{1 + m} ). Let me see if I can find similar expressions for ( q ) and ( r ).Starting again from the second equation: ( q = m(1 - r) ). From the third equation, ( r = m(1 - p) ). So, substituting ( r ) into the second equation:[q = m(1 - m(1 - p))][q = m - m^2(1 - p)][q = m - m^2 + m^2 p]But we already have ( p = frac{m}{1 + m} ), so substitute that in:[q = m - m^2 + m^2 left( frac{m}{1 + m} right )][q = m - m^2 + frac{m^3}{1 + m}]Let me combine these terms. To add them, I need a common denominator, which is ( 1 + m ):[q = frac{m(1 + m)}{1 + m} - frac{m^2(1 + m)}{1 + m} + frac{m^3}{1 + m}][q = frac{m + m^2 - m^2 - m^3 + m^3}{1 + m}][q = frac{m}{1 + m}]Wait, so ( q = frac{m}{1 + m} ) as well? That's the same as ( p ). But the problem states that ( p ), ( q ), and ( r ) are distinct complex numbers. So, if ( p = q ), that would contradict the given condition. Hmm, that suggests that my assumption might be leading to a contradiction unless ( m ) satisfies some condition.Let me check my steps again. When I substituted ( r = m(1 - p) ) into ( q = m(1 - r) ), I got:[q = m - m^2 + m^2 p]Then I substituted ( p = frac{m}{1 + m} ):[q = m - m^2 + m^2 left( frac{m}{1 + m} right )][q = m - m^2 + frac{m^3}{1 + m}]Then, combining terms:[q = frac{m(1 + m) - m^2(1 + m) + m^3}{1 + m}][= frac{m + m^2 - m^2 - m^3 + m^3}{1 + m}][= frac{m}{1 + m}]So, indeed, ( q = p ). But since ( p ), ( q ), ( r ) are distinct, this can't be. Therefore, my earlier assumption must have a flaw.Wait, perhaps I made a mistake in the substitution. Let me try a different approach. Instead of expressing ( p ) in terms of ( q ), ( q ) in terms of ( r ), and ( r ) in terms of ( p ), maybe I can set up a system of equations and solve for ( m ).Given:1. ( p = m(1 - q) )2. ( q = m(1 - r) )3. ( r = m(1 - p) )Let me substitute equation 3 into equation 2:( q = m(1 - r) = m(1 - m(1 - p)) )Then substitute equation 1 into this:( q = m(1 - m(1 - p)) = m(1 - m + m p) )But from equation 1, ( p = m(1 - q) ), so substitute that into the above:( q = m(1 - m + m cdot m(1 - q)) )( q = m(1 - m + m^2(1 - q)) )Expanding this:( q = m - m^2 + m^3(1 - q) )( q = m - m^2 + m^3 - m^3 q )Bring all terms with ( q ) to the left:( q + m^3 q = m - m^2 + m^3 )( q(1 + m^3) = m - m^2 + m^3 )( q = frac{m - m^2 + m^3}{1 + m^3} )Wait, this is the same as what I had for ( p ). So, ( q = p ). But that's a problem because ( p ), ( q ), ( r ) are supposed to be distinct. So, unless ( m ) is such that ( p ), ( q ), ( r ) are still distinct even if ( p = q ), which seems contradictory.Alternatively, maybe ( m ) is such that ( 1 + m^3 = 0 ), but then the denominator would be zero. So, if ( 1 + m^3 = 0 ), then ( m^3 = -1 ), so ( m = -1 ) or the other cube roots of -1. But let's see.Wait, in the equation ( q(1 + m^3) = m - m^2 + m^3 ), if ( 1 + m^3 = 0 ), then the left side is zero, so the right side must also be zero:( m - m^2 + m^3 = 0 )But ( m^3 = -1 ), so substitute:( m - m^2 - 1 = 0 )So, ( m^2 - m + 1 = 0 )Wait, that's a quadratic equation. Let me solve for ( m ):( m = frac{1 pm sqrt{1 - 4}}{2} = frac{1 pm sqrt{-3}}{2} = frac{1 pm isqrt{3}}{2} )So, these are the roots. So, if ( 1 + m^3 = 0 ), then ( m ) must satisfy ( m^2 - m + 1 = 0 ). But wait, ( m^3 = -1 ) implies ( m ) is a cube root of -1, which are ( -1 ), ( frac{1}{2} + ifrac{sqrt{3}}{2} ), and ( frac{1}{2} - ifrac{sqrt{3}}{2} ). But the solutions we just found are ( frac{1 pm isqrt{3}}{2} ), which are indeed the non-real cube roots of -1.So, in this case, if ( 1 + m^3 = 0 ), then ( m ) is one of these roots, and the equation ( q(1 + m^3) = m - m^2 + m^3 ) becomes ( 0 = 0 ), which is always true. So, in this case, ( p ), ( q ), ( r ) can be expressed in terms of ( m ), but we have to ensure they are distinct.Alternatively, if ( 1 + m^3 neq 0 ), then ( p = q ), which is not allowed. Therefore, the only possible values of ( m ) are the roots of ( m^2 - m + 1 = 0 ), which are ( frac{1 pm isqrt{3}}{2} ).So, these are the two possible values of ( m ). The problem asks for the product of all possible values of ( m ). Since these are roots of a quadratic equation ( m^2 - m + 1 = 0 ), the product of the roots is given by the constant term divided by the leading coefficient, which is ( 1 ).Therefore, the product of all possible values of ( m ) is ( 1 ).But wait, let me double-check. The quadratic equation is ( m^2 - m + 1 = 0 ), so the product is indeed ( 1 ). Alternatively, multiplying ( frac{1 + isqrt{3}}{2} ) and ( frac{1 - isqrt{3}}{2} ):[left( frac{1 + isqrt{3}}{2} right) left( frac{1 - isqrt{3}}{2} right) = frac{1 - (isqrt{3})^2}{4} = frac{1 - (-3)}{4} = frac{4}{4} = 1]Yes, that confirms it. So, the product is indeed ( 1 ).I think that's the answer. Let me just recap:- Expressed each variable in terms of the next.- Substituted recursively and found that unless ( 1 + m^3 = 0 ), ( p = q ), which is not allowed.- Solved ( 1 + m^3 = 0 ) leading to ( m^2 - m + 1 = 0 ).- Found the roots and their product is ( 1 ).So, I'm confident the product is ( 1 ).**Final Answer**The product of all possible values of ( m ) is boxed{1}.

💡Okay, so I have this problem about an artificial reservoir that's shaped like a rectangle. The sides differ by 1 km. There are two fishermen who start from the same corner and go to the diagonally opposite corner. One swims along the diagonal, and the other walks along the shore. Both move at 4 km/h, but one arrives 30 minutes earlier than the other. I need to find the dimensions of the reservoir.Alright, let's break this down step by step. First, let me visualize the rectangle. Let's say the shorter side is 'x' km, so the longer side would be 'x + 1' km because they differ by 1 km. That makes sense.Now, the first fisherman swims along the diagonal. To find the length of the diagonal, I can use the Pythagorean theorem. The diagonal 'd' would be the square root of (x squared plus (x + 1) squared). Let me write that down:d = √(x² + (x + 1)²)Simplifying that, it becomes √(x² + x² + 2x + 1) which is √(2x² + 2x + 1). Okay, got that.The second fisherman walks along the shore. Since it's a rectangle, he would walk along two sides. So, he walks the length of the shorter side and then the longer side. That would be x + (x + 1) = 2x + 1 km. Wait, actually, no. If he starts from one corner and goes to the opposite corner, he would walk along two sides, right? So, it's x + (x + 1) = 2x + 1 km. Yeah, that seems correct.Both fishermen are moving at 4 km/h. The one who swims along the diagonal will have a shorter distance but the same speed. However, the walking fisherman takes longer because he has to cover more distance. The problem says one arrives 30 minutes earlier. Since 30 minutes is 0.5 hours, I can set up an equation based on time.Time is equal to distance divided by speed. So, the time taken by the swimmer is √(2x² + 2x + 1) divided by 4. The time taken by the walker is (2x + 1) divided by 4. The difference between these times is 0.5 hours. So, the equation would be:(2x + 1)/4 - √(2x² + 2x + 1)/4 = 0.5Let me write that out:(2x + 1)/4 - √(2x² + 2x + 1)/4 = 0.5To simplify, I can multiply both sides by 4 to eliminate the denominators:2x + 1 - √(2x² + 2x + 1) = 2Now, let's isolate the square root term:2x + 1 - 2 = √(2x² + 2x + 1)Simplifying the left side:2x - 1 = √(2x² + 2x + 1)Now, to get rid of the square root, I'll square both sides:(2x - 1)² = (2x² + 2x + 1)Expanding the left side:4x² - 4x + 1 = 2x² + 2x + 1Now, let's bring all terms to one side:4x² - 4x + 1 - 2x² - 2x - 1 = 0Simplifying:2x² - 6x = 0Factor out a 2x:2x(x - 3) = 0So, the solutions are x = 0 or x = 3. Since x can't be 0 (as that wouldn't make sense for a reservoir), we have x = 3 km.Therefore, the shorter side is 3 km, and the longer side is x + 1 = 4 km.Let me double-check to make sure this makes sense. The diagonal would be √(3² + 4²) = 5 km. The swimmer's time is 5/4 = 1.25 hours. The walker's distance is 3 + 4 = 7 km, so his time is 7/4 = 1.75 hours. The difference is 0.5 hours, which is 30 minutes. That checks out.So, the dimensions of the reservoir are 3 km by 4 km.

💡First, I need to determine the total number of students in both arts and technical classes. There are 8 arts classes and 5 technical classes.Next, I'll assume that each class has the same number of students, denoted as ( n ).The total number of students in arts classes is ( 8n ), and in technical classes is ( 5n ).Now, I'll calculate the total sum of ages for each type of class. For arts classes, the total sum of ages is ( 8n times 21 ), and for technical classes, it's ( 5n times 18 ).Adding these together gives the overall total sum of ages for the university.Finally, I'll divide the total sum of ages by the total number of students to find the weighted average age of the university.