💡First, I recognize that ( x ) varies inversely as the square of ( y ), which means the relationship can be expressed as:[x = frac{k}{y^2}]where ( k ) is the constant of variation.Next, I use the given information that ( x = 1 ) for a certain value of ( y ). Plugging these values into the equation:[1 = frac{k}{y^2}]This implies that:[k = y^2]Then, I use the second piece of information that ( x = 0.1111111111111111 ) when ( y = 9 ) to find the constant ( k ):[0.1111111111111111 = frac{k}{9^2}][0.1111111111111111 = frac{k}{81}][k = 0.1111111111111111 times 81][k = 9]Now that I have determined ( k = 9 ), I can find the value of ( y ) when ( x = 1 ):[1 = frac{9}{y^2}][y^2 = 9][y = sqrt{9}][y = 3]Therefore, the value of ( y ) when ( x = 1 ) is 3.

💡First, I need to determine the distance each goat covers in the same amount of time. Gennadiy makes two jumps of 6 meters each, so he covers 12 meters. Nikolay makes three jumps of 4 meters each, also covering 12 meters. This means both goats have the same speed.Next, I need to calculate how many jumps each goat requires to reach the 2-kilometer mark. Nikolay's 4-meter jumps fit perfectly into 2000 meters, requiring exactly 500 jumps. Gennadiy's 6-meter jumps do not divide evenly into 2000 meters, so he needs to make 334 jumps to cover 2004 meters, which is slightly more than 2 kilometers.Since both goats have the same speed but Gennadiy has to travel a little extra distance, he will take more time to complete the journey. Therefore, Nikolay will finish the race faster.

💡To determine the probability of rain given that there is a strong wind, we can use the formula for conditional probability. First, we identify the given probabilities:- The probability of a strong wind, ( P(A) ), is 0.4.- The probability of rain, ( P(B) ), is 0.5.- The probability of both strong wind and rain, ( P(A cap B) ), is 0.3.Using the conditional probability formula:[P(B|A) = frac{P(A cap B)}{P(A)} = frac{0.3}{0.4} = frac{3}{4}]Therefore, the probability of rain under the condition of strong wind is (frac{3}{4}).

💡Okay, so I have this inequality to solve: (frac{x(x+2)}{(x-5)^2} ge 15). Hmm, let me try to figure out how to approach this. I remember that when dealing with inequalities involving rational expressions, I need to be careful about the denominator because it can't be zero, and also because multiplying both sides by a negative number would reverse the inequality sign. But in this case, the denominator is ((x-5)^2), which is always positive except when (x = 5), where it's undefined. So, I can multiply both sides by ((x-5)^2) without changing the direction of the inequality, as long as I remember that (x neq 5).Alright, so let me rewrite the inequality without the denominator by multiplying both sides by ((x-5)^2):[x(x + 2) ge 15(x - 5)^2]Now, I need to expand both sides to simplify this inequality. Let's start with the left side:[x(x + 2) = x^2 + 2x]And the right side:[15(x - 5)^2 = 15(x^2 - 10x + 25) = 15x^2 - 150x + 375]So, substituting these back into the inequality:[x^2 + 2x ge 15x^2 - 150x + 375]Now, I want to bring all terms to one side to set the inequality to zero. Let me subtract (x^2 + 2x) from both sides:[0 ge 15x^2 - 150x + 375 - x^2 - 2x]Simplify the right side by combining like terms:[0 ge 14x^2 - 152x + 375]Hmm, so the inequality is now:[14x^2 - 152x + 375 le 0]I need to solve this quadratic inequality. First, let me write it in standard form:[14x^2 - 152x + 375 le 0]To find the critical points where the expression equals zero, I can use the quadratic formula. The quadratic is (14x^2 - 152x + 375), so (a = 14), (b = -152), and (c = 375). Plugging these into the quadratic formula:[x = frac{-b pm sqrt{b^2 - 4ac}}{2a}]Calculating the discriminant first:[b^2 - 4ac = (-152)^2 - 4 times 14 times 375]Let me compute each part:[(-152)^2 = 23104][4 times 14 times 375 = 4 times 14 times 375 = 56 times 375 = 21000]So, the discriminant is:[23104 - 21000 = 2104]Wait, that doesn't seem right. Let me check my calculations again.Wait, ( (-152)^2 ) is indeed 23104. Then, 4 times 14 is 56, and 56 times 375. Let me compute 56 times 375:375 times 50 is 18,750, and 375 times 6 is 2,250. So, 18,750 + 2,250 = 21,000. So, the discriminant is 23,104 - 21,000 = 2,104.Wait, that's still 2,104. Hmm, okay, so the square root of 2,104. Let me see if that's a perfect square or not. Let me try to factor 2,104.Divide by 4: 2,104 ÷ 4 = 526. 526 ÷ 2 = 263. 263 is a prime number, I think. So, (sqrt{2104} = sqrt{4 times 526} = 2sqrt{526}). Hmm, that's not a nice number. Maybe I made a mistake earlier.Wait, let me double-check the discriminant calculation:[(-152)^2 = 152 times 152]Let me compute 150^2 = 22,500, and 2^2 = 4, and the cross term is 2*150*2 = 600. So, (150 + 2)^2 = 150^2 + 2*150*2 + 2^2 = 22,500 + 600 + 4 = 23,104. So that's correct.Then, 4ac = 4*14*375. Let's compute 14*375 first. 14*300=4,200 and 14*75=1,050, so total is 4,200 + 1,050 = 5,250. Then, 4ac = 4*5,250 = 21,000. So, discriminant is 23,104 - 21,000 = 2,104.So, (sqrt{2104}). Let me see if 46^2 is 2,116, which is more than 2,104. 45^2 is 2,025. So, 45^2 = 2,025, 46^2=2,116. So, 2,104 is between 45^2 and 46^2. Let me compute 45.8^2: 45 + 0.8. (45 + 0.8)^2 = 45^2 + 2*45*0.8 + 0.8^2 = 2,025 + 72 + 0.64 = 2,097.64. Hmm, still less than 2,104. 45.9^2: 45 + 0.9. (45 + 0.9)^2 = 2,025 + 2*45*0.9 + 0.81 = 2,025 + 81 + 0.81 = 2,106.81. So, 45.9^2 is 2,106.81, which is more than 2,104. So, sqrt(2104) is approximately 45.85.So, going back to the quadratic formula:[x = frac{152 pm 45.85}{28}]Wait, because b is -152, so -b is 152. So, the numerator is 152 ± sqrt(2104), which is approximately 152 ± 45.85.So, let's compute both roots:First root: (152 + 45.85)/28 ≈ (197.85)/28 ≈ 7.066Second root: (152 - 45.85)/28 ≈ (106.15)/28 ≈ 3.791Wait, so the roots are approximately 7.066 and 3.791.Wait, but earlier I thought they were 7.14 and 3.71, but maybe my approximation was off. Let me check with more precise calculations.Alternatively, maybe I can factor the quadratic, but given the coefficients, it might not factor nicely. So, perhaps it's better to use exact values.Wait, let me see if I can factor 14x^2 -152x +375. Let me try to factor it.Looking for two numbers that multiply to 14*375 = 5250 and add up to -152.Hmm, 5250 is a big number. Let me see, factors of 5250: 5250 = 50*105 = 50*15*7 = 2*25*15*7. Hmm, maybe 75 and 70: 75*70=5250, and 75 +70=145, which is not 152. Maybe 75 and 70, but with signs. Wait, since the middle term is -152, both numbers would be negative, so their sum would be negative, but their product is positive.Wait, 14x^2 -152x +375. Let me try to split the middle term:Looking for two numbers that multiply to 14*375=5250 and add up to -152.Let me try 75 and 70: 75 +70=145, which is close to 152 but not quite. Maybe 84 and 62.5? Wait, 84*62.5=5250, and 84+62.5=146.5, still not 152.Alternatively, maybe 105 and 50: 105*50=5250, and 105+50=155, which is close to 152 but not exact. Hmm, maybe it's not factorable, so I'll stick with the quadratic formula.So, the roots are approximately 3.791 and 7.066.Wait, so the quadratic (14x^2 -152x +375) can be written as (14(x - 3.791)(x - 7.066)). Since the coefficient of x^2 is positive, the parabola opens upwards. Therefore, the quadratic is less than or equal to zero between its roots.So, the solution to (14x^2 -152x +375 le 0) is (x in [3.791, 7.066]).But wait, I need to remember that in the original inequality, (x neq 5) because the denominator is zero there. So, I have to exclude x=5 from the solution.Therefore, the solution is (x in [3.791, 5) cup (5, 7.066]).But let me check if these approximate values are correct. Alternatively, maybe I can express the roots more precisely.Wait, let me compute the exact roots using the quadratic formula:[x = frac{152 pm sqrt{2104}}{28}]Simplify sqrt(2104). Let's factor 2104:2104 ÷ 4 = 526.526 ÷ 2 = 263, which is a prime number.So, sqrt(2104) = sqrt(4*526) = 2*sqrt(526).So, the roots are:[x = frac{152 pm 2sqrt{526}}{28} = frac{76 pm sqrt{526}}{14}]Hmm, that's exact form, but maybe I can write it as:[x = frac{76}{14} pm frac{sqrt{526}}{14} = frac{38}{7} pm frac{sqrt{526}}{14}]But that might not be necessary. Alternatively, I can write the roots as fractions:Wait, 152 divided by 28 simplifies to 38/7, which is approximately 5.4286.Similarly, sqrt(2104)/28 is sqrt(2104)/28. Since sqrt(2104) is approximately 45.86, so 45.86/28 ≈ 1.638.So, the roots are approximately 38/7 - 1.638 and 38/7 + 1.638.Wait, 38/7 is approximately 5.4286. So, 5.4286 - 1.638 ≈ 3.7906, and 5.4286 + 1.638 ≈ 7.0666.So, the roots are approximately 3.7906 and 7.0666.Therefore, the solution to the quadratic inequality is x between approximately 3.7906 and 7.0666, excluding x=5.But wait, in the original inequality, we had (frac{x(x+2)}{(x-5)^2} ge 15). So, the solution to this inequality is x in [3.7906, 5) union (5, 7.0666].But let me test a value in each interval to ensure that the inequality holds.The critical points are x ≈ 3.7906, x=5, and x≈7.0666.So, the intervals are:1. (-∞, 3.7906)2. (3.7906, 5)3. (5, 7.0666)4. (7.0666, ∞)Let me pick a test point in each interval.1. Let's pick x=0 in (-∞, 3.7906):Compute (frac{0(0+2)}{(0-5)^2} = 0). Is 0 ≥15? No. So, this interval is not part of the solution.2. Let's pick x=4 in (3.7906,5):Compute (frac{4(4+2)}{(4-5)^2} = frac{4*6}{1} = 24). Is 24 ≥15? Yes. So, this interval is part of the solution.3. Let's pick x=6 in (5,7.0666):Compute (frac{6(6+2)}{(6-5)^2} = frac{6*8}{1} = 48). Is 48 ≥15? Yes. So, this interval is part of the solution.4. Let's pick x=8 in (7.0666, ∞):Compute (frac{8(8+2)}{(8-5)^2} = frac{8*10}{9} ≈ 8.888). Is 8.888 ≥15? No. So, this interval is not part of the solution.Therefore, the solution is x in [3.7906,5) union (5,7.0666].But wait, I should check the endpoints. At x≈3.7906 and x≈7.0666, the expression equals 15, so they should be included.But I need to write the exact values, not the approximate decimals. So, let me express the roots exactly.From earlier, the roots are:[x = frac{76 pm sqrt{526}}{14}]Simplify:[x = frac{76}{14} pm frac{sqrt{526}}{14} = frac{38}{7} pm frac{sqrt{526}}{14}]But perhaps it's better to write them as:[x = frac{76 pm sqrt{2104}}{28}]But 2104 = 4*526, so sqrt(2104)=2*sqrt(526). So, the roots are:[x = frac{76 pm 2sqrt{526}}{28} = frac{38 pm sqrt{526}}{14}]So, the exact roots are (frac{38 - sqrt{526}}{14}) and (frac{38 + sqrt{526}}{14}).Therefore, the solution in exact terms is:[x in left[ frac{38 - sqrt{526}}{14}, 5 right) cup left(5, frac{38 + sqrt{526}}{14} right]]But let me check if I can simplify (frac{38}{14}). 38 divided by 14 is 19/7, so:[x = frac{19}{7} pm frac{sqrt{526}}{14}]Alternatively, combining the terms:[x = frac{19 pm frac{sqrt{526}}{2}}{7}]But perhaps it's better to leave it as (frac{38 pm sqrt{526}}{14}).Wait, but 38 and 14 have a common factor of 2, so:[x = frac{19 pm frac{sqrt{526}}{2}}{7}]Hmm, maybe not necessary. Alternatively, I can write it as:[x = frac{38 pm sqrt{526}}{14}]Which is fine.So, the solution is:[x in left[ frac{38 - sqrt{526}}{14}, 5 right) cup left(5, frac{38 + sqrt{526}}{14} right]]But let me verify if these exact roots are correct.Wait, let me plug x=3.7906 back into the original inequality:[frac{3.7906*(3.7906 + 2)}{(3.7906 -5)^2} = frac{3.7906*5.7906}{( -1.2094)^2} ≈ frac{21.95}{1.4627} ≈ 15.00]Similarly, for x=7.0666:[frac{7.0666*(7.0666 + 2)}{(7.0666 -5)^2} = frac{7.0666*9.0666}{(2.0666)^2} ≈ frac{64.06}{4.27} ≈ 15.00]So, the endpoints are indeed where the expression equals 15, so they should be included.Therefore, the solution is x in [ (38 - sqrt(526))/14, 5 ) union (5, (38 + sqrt(526))/14 ].But to write this in interval notation, I can express it as:[left[ frac{38 - sqrt{526}}{14}, 5 right) cup left(5, frac{38 + sqrt{526}}{14} right]]Alternatively, since 38/14 simplifies to 19/7, and sqrt(526)/14 is as simplified as it gets, I can write:[left[ frac{19}{7} - frac{sqrt{526}}{14}, 5 right) cup left(5, frac{19}{7} + frac{sqrt{526}}{14} right]]But perhaps it's better to rationalize or write it differently, but I think this is acceptable.Wait, let me check if I can write sqrt(526) as something else. 526 factors into 2*263, and 263 is a prime number, so sqrt(526) cannot be simplified further.Therefore, the exact solution is:[x in left[ frac{38 - sqrt{526}}{14}, 5 right) cup left(5, frac{38 + sqrt{526}}{14} right]]Alternatively, I can write this as:[x in left[ frac{19 - frac{sqrt{526}}{2}}{7}, 5 right) cup left(5, frac{19 + frac{sqrt{526}}{2}}{7} right]]But I think the first form is clearer.So, to summarize, the solution to the inequality (frac{x(x+2)}{(x-5)^2} ge 15) is all real numbers x such that x is between (frac{38 - sqrt{526}}{14}) and 5, excluding 5, and between 5 and (frac{38 + sqrt{526}}{14}), excluding 5.Therefore, in interval notation, the solution is:[left[ frac{38 - sqrt{526}}{14}, 5 right) cup left(5, frac{38 + sqrt{526}}{14} right]]I think that's the final answer.

💡To solve the problem, we need to determine the point where the mouse starts getting farther from the cheese. The mouse is running along the line ( y = -5x + 18 ), and the cheese is located at ( (12, 10) ).1. **Identify the line on which the mouse is running:** The mouse is running up the line given by the equation ( y = -5x + 18 ).2. **Find the slope of the line perpendicular to ( y = -5x + 18 ):** The slope of the given line is (-5). The slope of a line perpendicular to this line is the negative reciprocal of (-5), which is (frac{1}{5}).3. **Write the equation of the line passing through the cheese and perpendicular to ( y = -5x + 18 ):** The cheese is located at ((12, 10)). Using the point-slope form of the equation of a line, we have: [ y - 10 = frac{1}{5}(x - 12) ] Simplifying this, we get: [ y - 10 = frac{1}{5}x - frac{12}{5} ] [ y = frac{1}{5}x + 10 - frac{12}{5} ] [ y = frac{1}{5}x + frac{50}{5} - frac{12}{5} ] [ y = frac{1}{5}x + frac{38}{5} ]4. **Find the intersection of the two lines:** We need to solve the system of equations: [ y = -5x + 18 ] [ y = frac{1}{5}x + frac{38}{5} ] Setting the two equations equal to each other: [ -5x + 18 = frac{1}{5}x + frac{38}{5} ] To clear the fraction, multiply every term by 5: [ -25x + 90 = x + 38 ] Combine like terms: [ -25x - x = 38 - 90 ] [ -26x = -52 ] Solving for ( x ): [ x = 2 ]5. **Find the corresponding ( y )-coordinate:** Substitute ( x = 2 ) back into the equation ( y = -5x + 18 ): [ y = -5(2) + 18 ] [ y = -10 + 18 ] [ y = 8 ]6. **Calculate ( a + b ):** The point where the mouse starts getting farther from the cheese is ((2, 8)). Therefore, ( a + b = 2 + 8 = 10 ).The final answer is ( boxed{10} ).

💡First, I need to determine how many candies Lillian had initially before her father gave her additional candies.I know that after receiving 5 more candies, Lillian has a total of 93 candies.To find the initial number of candies, I can subtract the 5 candies her father gave her from the total number of candies she has now.So, 93 candies (total after receiving 5 more) minus 5 candies (given by her father) equals 88 candies.Therefore, Lillian initially collected 88 candies.