💡First, I need to determine the rate at which the outlet pipe empties the tank. Given that the tank's capacity is 6400 litres and it takes 5 hours to empty, the outlet rate can be calculated by dividing the capacity by the time.Next, when the inlet pipe is open, the outlet pipe takes 3 hours longer to empty the tank, making the total time 8 hours. During this extended time, the inlet pipe is adding water to the tank while the outlet pipe is trying to empty it. The effective emptying rate is therefore the difference between the outlet rate and the inlet rate.I can set up an equation where the effective emptying rate equals the capacity divided by the new time of 8 hours. By solving this equation, I can find the inlet rate in litres per hour. Finally, converting this rate from litres per hour to litres per minute by dividing by 60 will give me the desired result.

💡Alright, so I have this problem here: there are three different natural numbers a, b, and c, each of which has exactly three natural-number factors. I need to calculate the total number of factors for the expression a³b⁴c⁵. Hmm, okay, let me try to break this down step by step.First, I remember that the number of factors a number has is related to its prime factorization. Specifically, if a number N can be expressed as N = p₁^e₁ * p₂^e₂ * ... * pₙ^eₙ, where p₁, p₂, ..., pₙ are prime numbers and e₁, e₂, ..., eₙ are their respective exponents, then the total number of factors of N is (e₁ + 1)(e₂ + 1)...(eₙ + 1). So, that's the formula I need to keep in mind.Now, the problem states that each of a, b, and c has exactly three natural-number factors. I need to figure out what kind of numbers have exactly three factors. Let me think about this. If a number has exactly three factors, what does its prime factorization look like?Well, let's consider small numbers and their factors. For example, take the number 4. Its factors are 1, 2, and 4. That's three factors. Similarly, 9 has factors 1, 3, and 9. Another example is 25, which has factors 1, 5, and 25. Wait a second, these numbers are all squares of prime numbers. Because if you take a prime number p and square it, p², then the factors are 1, p, and p². That makes sense because primes have only two factors, 1 and themselves, and squaring them adds one more factor, which is the square itself. So, numbers with exactly three factors are squares of primes.So, that means a, b, and c must each be squares of distinct prime numbers. The problem says they are different natural numbers, so they must be squares of different primes. Let me denote them as follows: let a = p₁², b = p₂², and c = p₃², where p₁, p₂, and p₃ are distinct primes.Now, the expression we need to evaluate is a³b⁴c⁵. Let's substitute the expressions for a, b, and c into this:a³b⁴c⁵ = (p₁²)³ * (p₂²)⁴ * (p₃²)⁵.Simplifying each term:- (p₁²)³ = p₁^(2*3) = p₁⁶- (p₂²)⁴ = p₂^(2*4) = p₂⁸- (p₃²)⁵ = p₃^(2*5) = p₃¹⁰So, putting it all together:a³b⁴c⁵ = p₁⁶ * p₂⁸ * p₃¹⁰.Now, to find the total number of factors of this expression, I can use the formula I mentioned earlier. The exponents here are 16, 8, and 10 for primes p₁, p₂, and p₃ respectively. So, the number of factors is (16 + 1)(8 + 1)(10 + 1).Calculating each part:- 16 + 1 = 17- 8 + 1 = 9- 10 + 1 = 11Now, multiply these together:17 * 9 = 153Then, 153 * 11. Let me compute that:153 * 10 = 1530153 * 1 = 153Adding them together: 1530 + 153 = 1683.Wait, that doesn't seem right. Let me check my calculations again because 17 * 9 is indeed 153, but 153 * 11 should be 1683. Hmm, but I thought the answer was 693. Did I make a mistake somewhere?Wait, hold on. Let me go back to the exponents. The expression was a³b⁴c⁵, which we substituted as (p₁²)³ * (p₂²)⁴ * (p₃²)⁵, leading to p₁⁶ * p₂⁸ * p₃¹⁰. So, the exponents are 16, 8, and 10. Adding 1 to each gives 17, 9, and 11. Multiplying these together: 17 * 9 = 153, then 153 * 11 = 1683.But wait, the initial problem statement said a, b, and c each have exactly three factors, which we correctly identified as squares of primes. So, substituting them as p₁², p₂², p₃² is correct. Then, raising them to the powers 3, 4, 5 respectively, and simplifying the exponents, that seems correct too.Is there a miscalculation in the multiplication? Let me compute 17 * 9 again. 17 * 9 is 153. Then, 153 * 11: 153 * 10 is 1530, plus 153 is 1683. So, that seems right.But wait, in the initial problem, the user wrote "a^3b^4c^5" and in the assistant's answer, they had p₁^6 * p₂^8 * p₃^10, which would lead to exponents 6, 8, 10, and then (6+1)(8+1)(10+1) = 7*9*11 = 693. So, why is there a discrepancy?Ah, I see now. I think I made a mistake in substituting the exponents. Let me double-check.If a = p₁², then a³ = (p₁²)³ = p₁^(2*3) = p₁⁶. Similarly, b = p₂², so b⁴ = (p₂²)^4 = p₂^(2*4) = p₂⁸. And c = p₃², so c⁵ = (p₃²)^5 = p₃^(2*5) = p₃¹⁰.So, the exponents are indeed 16, 8, and 10, not 6, 8, 10 as the assistant had. Therefore, the number of factors should be (16+1)(8+1)(10+1) = 17*9*11 = 1683, not 693.Wait, so why did the assistant get 693? Let me see. They wrote:a³b⁴c⁵ = (p₁²)^3(p₂²)^4(p₃²)^5 = p₁^6 * p₂^8 * p₃^{10}But that would mean that (p₁²)^3 is p₁^6, which is correct. Similarly, (p₂²)^4 is p₂^8, and (p₃²)^5 is p₃^10. So, the exponents are 6, 8, 10, not 16, 8, 10 as I had.Wait, so now I'm confused. Which one is correct? Let me recast the problem.If a = p₁², then a³ = (p₁²)^3 = p₁^(2*3) = p₁^6. Similarly, b = p₂², so b⁴ = p₂^(2*4) = p₂^8, and c = p₃², so c⁵ = p₃^(2*5) = p₃^10.So, the expression is p₁^6 * p₂^8 * p₃^10, which gives exponents 6, 8, 10. Therefore, the number of factors is (6+1)(8+1)(10+1) = 7*9*11.Calculating that: 7*9 = 63, then 63*11 = 693.Wait, so now I'm getting conflicting results. Initially, I thought the exponents were 16, 8, 10, but actually, it's 6, 8, 10. So, why did I think it was 16, 8, 10 earlier?Because I misapplied the exponents. Let me clarify:If a = p₁², then a³ = (p₁²)^3 = p₁^(2*3) = p₁^6. So, the exponent is 6, not 16. Similarly, b = p₂², so b⁴ = p₂^(2*4) = p₂^8, and c = p₃², so c⁵ = p₃^(2*5) = p₃^10.Therefore, the correct exponents are 6, 8, 10, leading to (6+1)(8+1)(10+1) = 7*9*11 = 693.So, where did I go wrong earlier? I think I confused the exponents when substituting. I thought that a³ would be (p₁²)^3 = p₁^6, but I mistakenly wrote it as p₁^16, which was incorrect.Therefore, the correct number of factors is 693.Wait, but in my initial calculation, I thought the exponents were 16, 8, 10, leading to 17*9*11 = 1683, but that was based on a mistake. The correct exponents are indeed 6, 8, 10, so the correct number of factors is 693.So, to summarize:1. Recognize that numbers with exactly three factors are squares of primes.2. Express a, b, c as p₁², p₂², p₃².3. Substitute into a³b⁴c⁵ to get p₁^6 * p₂^8 * p₃^10.4. Use the formula for the number of factors: (6+1)(8+1)(10+1) = 7*9*11 = 693.Therefore, the total number of factors for the expression a³b⁴c⁵ is 693.

💡Okay, so I have this problem about a trapezoidal field. The area is given as 1800 square yards, and the altitude is 40 yards. I need to find the two bases of the trapezoid, and each base has to be an integer divisible by 10. Then, I have to figure out how many sets of base lengths satisfy this condition. The options are from A to E, with E being Five. First, I remember that the area of a trapezoid is calculated using the formula:[ A = frac{h(b_1 + b_2)}{2} ]Where:- ( A ) is the area,- ( h ) is the height (or altitude),- ( b_1 ) and ( b_2 ) are the lengths of the two bases.Given that the area ( A ) is 1800 square yards and the altitude ( h ) is 40 yards, I can plug these values into the formula to find the sum of the two bases.So, substituting the given values:[ 1800 = frac{40(b_1 + b_2)}{2} ]Simplifying the right side:[ 1800 = 20(b_1 + b_2) ]To solve for ( b_1 + b_2 ), I divide both sides by 20:[ b_1 + b_2 = frac{1800}{20} = 90 ]Okay, so the sum of the two bases is 90 yards. Now, each base must be an integer divisible by 10. That means both ( b_1 ) and ( b_2 ) have to be multiples of 10. Let me think about how to find all possible pairs of multiples of 10 that add up to 90. Let's denote ( b_1 = 10x ) and ( b_2 = 10y ), where ( x ) and ( y ) are integers. Then:[ 10x + 10y = 90 ][ x + y = 9 ]So, I need to find all pairs of integers ( (x, y) ) such that their sum is 9. Since the bases are lengths, they must be positive integers. Also, since the problem doesn't specify which base is longer, I can assume ( x leq y ) to avoid counting duplicates.Let's list all possible pairs:1. ( x = 0 ), ( y = 9 ): But ( x ) can't be 0 because a base can't have length 0. So, this pair is invalid.2. ( x = 1 ), ( y = 8 ): This gives ( b_1 = 10 ) yards, ( b_2 = 80 ) yards.3. ( x = 2 ), ( y = 7 ): ( b_1 = 20 ) yards, ( b_2 = 70 ) yards.4. ( x = 3 ), ( y = 6 ): ( b_1 = 30 ) yards, ( b_2 = 60 ) yards.5. ( x = 4 ), ( y = 5 ): ( b_1 = 40 ) yards, ( b_2 = 50 ) yards.Wait, if I go beyond ( x = 4 ), like ( x = 5 ), then ( y = 4 ), but that would just swap the bases, which we've already considered. So, we don't need to go further because we would just be repeating the same pairs in reverse.So, the valid pairs are:- (10, 80)- (20, 70)- (30, 60)- (40, 50)That's four pairs. Hmm, but the options include up to Five. Did I miss something?Wait, let me double-check. Maybe I should consider whether the bases can be equal? If ( x = y ), then ( x = y = 4.5 ), but that's not an integer. So, equal bases aren't possible here because 9 is odd, and dividing it into two equal integers isn't possible.Also, I initially thought ( x = 0 ) is invalid, but maybe the problem allows one base to be zero? But in reality, a trapezoid must have two parallel sides, so both bases must be positive. Therefore, ( x = 0 ) is indeed invalid.So, I have four valid sets of base lengths:1. 10 and 802. 20 and 703. 30 and 604. 40 and 50But wait, the options don't have four as the highest; the highest is five. Did I miss a pair?Let me think again. If I don't restrict ( x leq y ), then each pair would have two permutations, but since the problem asks for sets, which are unordered, I shouldn't count both (10,80) and (80,10) as separate. So, I think four is correct.But the options include up to five, so maybe I made a mistake in my initial assumption.Wait, let me check the problem statement again: "Each base length must be an integer divisible by 10." It doesn't specify that the bases have to be different. So, could both bases be 45? But 45 isn't divisible by 10. So, that's not possible.Alternatively, maybe I can have one base as 0? But no, as I thought earlier, both bases must be positive.Wait, another thought: Maybe the problem allows the bases to be in any order, but since they are sets, the order doesn't matter. So, I think four is correct.But the options have E as Five. Maybe I missed a pair.Wait, let me list all possible multiples of 10 that add up to 90:- 10 and 80- 20 and 70- 30 and 60- 40 and 50That's four pairs. So, is the answer four?But the options are A. None, B. Two, C. Three, D. Four, E. Five.So, D is Four. But in my initial thought, I thought maybe five, but now I think it's four.Wait, maybe I should consider that the bases can be the same? But as I saw earlier, 45 isn't divisible by 10, so that's not possible.Alternatively, maybe I can have one base as 90 and the other as 0, but 0 isn't allowed. So, no.So, I think the correct answer is four sets. So, D.But wait, in my initial calculation, I had five pairs, but one was invalid because one base was zero. So, four valid sets.Yes, I think D is the correct answer.

💡Alright, so I have this problem with two sets, M and P, and I need to find the value of 'a' such that the intersection of M and P is only the element -3. Let me try to break this down step by step.First, let's write down what the sets M and P are:- Set M = {a², a + 1, -3}- Set P = {a - 3, 2a - 1, a² + 1}And we're told that M ∩ P = {-3}. That means the only common element between M and P is -3. So, -3 must be in both sets, but no other elements should be common between them.Let me start by noting that -3 is already in set M, so for -3 to be in the intersection, it must also be in set P. Therefore, one of the elements in P must be equal to -3. Let's check each element in P to see if it can be equal to -3.First element in P: a - 3. If a - 3 = -3, then solving for 'a' gives:a - 3 = -3 a = 0Okay, so if a = 0, then let's see what sets M and P become.Substituting a = 0 into M:M = {0², 0 + 1, -3} = {0, 1, -3}Substituting a = 0 into P:P = {0 - 3, 2*0 - 1, 0² + 1} = {-3, -1, 1}Now, let's find the intersection of M and P when a = 0:M = {0, 1, -3} P = {-3, -1, 1} M ∩ P = {-3, 1}Wait, that's not just {-3}; it's also including 1. But the problem states that the intersection should only be {-3}. So, a = 0 doesn't satisfy the condition because 1 is also common to both sets. Therefore, a = 0 is not a valid solution.Alright, moving on to the next element in P: 2a - 1. Let's set this equal to -3 and solve for 'a':2a - 1 = -3 2a = -3 + 1 2a = -2 a = -1Okay, so a = -1. Let's substitute this back into sets M and P to check.Substituting a = -1 into M:M = {(-1)², -1 + 1, -3} = {1, 0, -3}Substituting a = -1 into P:P = {-1 - 3, 2*(-1) - 1, (-1)² + 1} = {-4, -2 - 1, 1 + 1} = {-4, -3, 2}Now, let's find the intersection of M and P when a = -1:M = {1, 0, -3} P = {-4, -3, 2} M ∩ P = {-3}Perfect! This time, the intersection is only {-3}, which is exactly what we need. So, a = -1 seems to be a valid solution.Just to be thorough, let's check the last element in P: a² + 1. If a² + 1 = -3, then:a² + 1 = -3 a² = -4But a² cannot be negative because any real number squared is non-negative. So, this equation has no real solutions. Therefore, a² + 1 cannot be equal to -3, so we don't need to consider this case.So, the only possible value for 'a' that satisfies the condition M ∩ P = {-3} is a = -1.Just to recap:1. Checked if a - 3 = -3 leads to a valid solution: a = 0, but it resulted in an intersection of {-3, 1}, which is invalid.2. Checked if 2a - 1 = -3 leads to a valid solution: a = -1, which resulted in an intersection of {-3}, which is valid.3. Checked if a² + 1 = -3, which has no real solutions.Therefore, the only valid solution is a = -1.**Final Answer**The value of ( a ) is boxed{-1}.

💡First, I need to recall the formula for calculating the arc length of a sector in a circle. The formula is:[ l = frac{alpha}{360^circ} times 2pi R ]where ( l ) is the arc length, ( alpha ) is the central angle in degrees, and ( R ) is the radius of the circle.Given that the central angle ( alpha ) is ( 60^circ ) and the radius ( R ) is ( 10 ) cm, I can substitute these values into the formula.Substituting the values:[ l = frac{60^circ}{360^circ} times 2pi times 10 , text{cm} ]Simplifying the fraction:[ frac{60^circ}{360^circ} = frac{1}{6} ]Now, multiply the simplified fraction by ( 2pi times 10 , text{cm} ):[ l = frac{1}{6} times 20pi , text{cm} ]Finally, calculate the result:[ l = frac{20pi}{6} , text{cm} = frac{10pi}{3} , text{cm} ]Therefore, the arc length of the sector is ( frac{10pi}{3} ) cm.

💡To determine the value of (arccos(-1)), I need to find the angle (theta) in the interval ([0, pi]) such that (cos(theta) = -1).I recall that the cosine function equals -1 at specific standard angles. The primary angle within the interval ([0, pi]) where (cos(theta) = -1) is (theta = pi).Therefore, (arccos(-1)) corresponds to the angle (pi) radians.