answer：First, I need to determine the total number of popcorn kernels Jenny dropped on her way home. She lives 5000 feet away from her school and drops 1 kernel of popcorn every 25 feet. To find the total number of kernels dropped, I divide the total distance by the distance per kernel:5000 feet ÷ 25 feet/kernel = 200 kernels.Next, a squirrel ate one-quarter of the popcorn Jenny dropped. To find out how many kernels were eaten, I calculate one-quarter of 200:200 kernels × 1/4 = 50 kernels.Finally, to find the number of popcorn kernels remaining on the ground, I subtract the number eaten by the squirrel from the total number dropped:200 kernels - 50 kernels = 150 kernels.Therefore, Jenny will find 150 popcorn kernels remaining on the ground the following morning.

answer：Okay, so I have this problem where I start with T₁ = 2, and then each subsequent term is defined by Tₙ₊₁ = Tₙ² - Tₙ + 1. I need to prove two things: first, that all the terms in the sequence are pairwise relatively prime, and second, that the sum of 1/Tₖ from k=1 to n approaches 1 as n goes to infinity. Hmm, that sounds interesting. Let me try to break this down step by step.Starting with the first part: proving that the terms are pairwise relatively prime. That means that for any two different terms Tᵢ and Tⱼ, their greatest common divisor (gcd) is 1. So, I need to show that any two terms in the sequence don't share any common factors other than 1.Let me write out the first few terms to see if I can spot a pattern or get some intuition.Given T₁ = 2.Then T₂ = T₁² - T₁ + 1 = 2² - 2 + 1 = 4 - 2 + 1 = 3.Next, T₃ = T₂² - T₂ + 1 = 3² - 3 + 1 = 9 - 3 + 1 = 7.Then T₄ = T₃² - T₃ + 1 = 7² - 7 + 1 = 49 - 7 + 1 = 43.T₅ would be 43² - 43 + 1 = 1849 - 43 + 1 = 1807.Hmm, so the terms are 2, 3, 7, 43, 1807, and so on. Looking at these, they seem to be prime numbers, but not necessarily. For example, 1807 is 13 × 139, so it's not prime. But still, each term is relatively prime to the previous ones.Wait, let me check the gcd between T₁ and T₂: gcd(2, 3) = 1. Between T₂ and T₃: gcd(3, 7) = 1. Between T₃ and T₄: gcd(7, 43) = 1. Between T₄ and T₅: gcd(43, 1807). Since 1807 ÷ 43 is 42.023... Wait, 43 × 42 = 1806, so 1807 - 1806 = 1. So 1807 = 43 × 42 + 1, which means 43 and 1807 are coprime. So, yes, gcd(43, 1807) = 1.So, it seems that each term is coprime with the next one. But I need to show that all pairs are coprime, not just consecutive ones. For example, is gcd(T₁, T₃) = gcd(2, 7) = 1? Yes. gcd(T₁, T₄) = gcd(2, 43) = 1. Similarly, gcd(T₂, T₄) = gcd(3, 43) = 1. So, all pairs so far are coprime. That's a good sign.To prove this in general, maybe I can use induction or some property of the recurrence relation. Let me think about the recurrence: Tₙ₊₁ = Tₙ² - Tₙ + 1. Maybe I can express this in a way that relates Tₙ₊₁ and Tₙ, and then see how it affects the gcd.Suppose I have two terms Tₖ and Tⱼ where k < j. I need to show that gcd(Tₖ, Tⱼ) = 1. Let's assume that for all m < j, gcd(Tₖ, Tₘ) = 1. Then, if I can show that gcd(Tₖ, Tⱼ) = 1, that would complete the induction step.Alternatively, maybe I can use the property that if d divides Tₖ and Tⱼ, then d divides some combination of them. Let me try to see if I can find a relationship between Tₙ and Tₙ₊₁.From the recurrence, Tₙ₊₁ = Tₙ² - Tₙ + 1. Let's rearrange this: Tₙ₊₁ - 1 = Tₙ(Tₙ - 1). So, Tₙ divides Tₙ₊₁ - 1. That is, Tₙ | (Tₙ₊₁ - 1). So, Tₙ₊₁ ≡ 1 mod Tₙ.Similarly, Tₙ₊₂ = Tₙ₊₁² - Tₙ₊₁ + 1. Let's compute Tₙ₊₂ mod Tₙ. Since Tₙ₊₁ ≡ 1 mod Tₙ, then Tₙ₊₁² ≡ 1² = 1 mod Tₙ, and Tₙ₊₁ ≡ 1 mod Tₙ. Therefore, Tₙ₊₂ ≡ 1 - 1 + 1 = 1 mod Tₙ. So, Tₙ₊₂ ≡ 1 mod Tₙ.Wait, so Tₙ₊₁ ≡ 1 mod Tₙ, and Tₙ₊₂ ≡ 1 mod Tₙ. Similarly, Tₙ₊₃ = Tₙ₊₂² - Tₙ₊₂ + 1. Since Tₙ₊₂ ≡ 1 mod Tₙ, then Tₙ₊₃ ≡ 1² - 1 + 1 = 1 mod Tₙ. So, inductively, all terms after Tₙ₊₁ are congruent to 1 mod Tₙ.Therefore, for any m > n, Tₘ ≡ 1 mod Tₙ. So, Tₘ = k*Tₙ + 1 for some integer k. Therefore, if d divides both Tₙ and Tₘ, then d divides Tₘ - k*Tₙ = 1. So, d divides 1, which means d = 1. Therefore, gcd(Tₙ, Tₘ) = 1 for any m > n.That's a solid argument. So, since for any two terms where one comes after the other, their gcd is 1, and since the sequence is built in such a way that each term is coprime with all previous terms, the entire sequence is pairwise coprime. That takes care of the first part.Now, moving on to the second part: proving that the sum of 1/Tₖ from k=1 to infinity converges to 1. Wait, actually, the problem says "the limit of the following series is 1," but it's written as a finite sum up to n. So, I think it's asking to show that as n approaches infinity, the sum approaches 1. So, the infinite series converges to 1.Let me write out the partial sums:S_n = 1/T₁ + 1/T₂ + 1/T₃ + ... + 1/Tₙ.We need to show that as n approaches infinity, S_n approaches 1.Looking at the terms, they grow very rapidly. The first few terms are 2, 3, 7, 43, 1807, etc. So, 1/2 + 1/3 + 1/7 + 1/43 + 1/1807 + ... This is a rapidly converging series because each term is much smaller than the previous one.But how do we show that the sum converges to exactly 1? Maybe there's a telescoping series or some recursive relationship that can help us express the sum in a way that telescopes.Let me look back at the recurrence relation: Tₙ₊₁ = Tₙ² - Tₙ + 1. Let's try to manipulate this equation to find a relationship involving 1/Tₙ.Rearranging the recurrence: Tₙ₊₁ - 1 = Tₙ² - Tₙ = Tₙ(Tₙ - 1). So, Tₙ₊₁ - 1 = Tₙ(Tₙ - 1). Therefore, 1/(Tₙ - 1) = Tₙ / (Tₙ₊₁ - 1). Hmm, not sure if that helps directly.Wait, let's consider 1/(Tₙ - 1) - 1/Tₙ. Let's compute that:1/(Tₙ - 1) - 1/Tₙ = [Tₙ - (Tₙ - 1)] / [Tₙ(Tₙ - 1)] = [1] / [Tₙ(Tₙ - 1)].But from the recurrence, Tₙ₊₁ - 1 = Tₙ(Tₙ - 1). So, 1/(Tₙ - 1) - 1/Tₙ = 1/(Tₙ₊₁ - 1).Therefore, 1/(Tₙ - 1) - 1/Tₙ = 1/(Tₙ₊₁ - 1).That's interesting. Let me write that down:1/(Tₙ - 1) - 1/Tₙ = 1/(Tₙ₊₁ - 1).So, rearranging, we get:1/(Tₙ - 1) - 1/(Tₙ₊₁ - 1) = 1/Tₙ.Therefore, 1/Tₙ = 1/(Tₙ - 1) - 1/(Tₙ₊₁ - 1).That's a telescoping difference! So, each term 1/Tₙ can be expressed as the difference between two terms, which suggests that when we sum them up, the series telescopes.Let me write out the partial sum S_n:S_n = sum_{k=1}^n 1/T_k = sum_{k=1}^n [1/(T_k - 1) - 1/(T_{k+1} - 1)].This is a telescoping series, so when we expand it, most terms cancel out:= [1/(T₁ - 1) - 1/(T₂ - 1)] + [1/(T₂ - 1) - 1/(T₃ - 1)] + ... + [1/(T_n - 1) - 1/(T_{n+1} - 1)].All the intermediate terms cancel, leaving:S_n = 1/(T₁ - 1) - 1/(T_{n+1} - 1).Given that T₁ = 2, so T₁ - 1 = 1. Therefore, 1/(T₁ - 1) = 1/1 = 1.So, S_n = 1 - 1/(T_{n+1} - 1).Now, as n approaches infinity, T_{n+1} grows very rapidly. From the recurrence, each term is roughly the square of the previous term, so T_{n} grows doubly exponentially. Therefore, 1/(T_{n+1} - 1) approaches 0 as n approaches infinity.Thus, the limit of S_n as n approaches infinity is:lim_{n→∞} S_n = 1 - 0 = 1.Therefore, the sum converges to 1.Let me double-check this reasoning. We expressed each 1/Tₖ as a telescoping difference, summed them up, and saw that the partial sum simplifies to 1 minus a term that goes to zero. That seems correct.Just to be thorough, let's compute the partial sums numerically for the first few terms and see if they approach 1.Compute S₁ = 1/2 = 0.5S₂ = 1/2 + 1/3 ≈ 0.5 + 0.3333 ≈ 0.8333S₃ = 0.8333 + 1/7 ≈ 0.8333 + 0.1429 ≈ 0.9762S₄ = 0.9762 + 1/43 ≈ 0.9762 + 0.02326 ≈ 0.99946S₅ = 0.99946 + 1/1807 ≈ 0.99946 + 0.000553 ≈ 0.999999...So, indeed, the partial sums are approaching 1 very quickly. After just five terms, we're already at about 0.999999, which is extremely close to 1. This numerical evidence supports our analytical result.Therefore, both parts of the problem are proven: the terms are pairwise relatively prime, and the sum of their reciprocals converges to 1.**Final Answer**The terms of the sequence are pairwise relatively prime, and the limit of the series is boxed{1}.

answer：Okay, so I have this problem where there are two positive integers, a and b. Their difference is 120, and the least common multiple (LCM) of them is 105 times their greatest common divisor (GCD). I need to find the larger number between a and b. Hmm, let me think about how to approach this.First, I remember that for any two numbers, the product of their LCM and GCD is equal to the product of the numbers themselves. So, that is, LCM(a, b) * GCD(a, b) = a * b. Maybe that can help me here.Given that the LCM is 105 times the GCD, I can write that as LCM(a, b) = 105 * GCD(a, b). Let me denote the GCD as d. So, GCD(a, b) = d. Then, the LCM would be 105d.Since d is the GCD of a and b, I can express a and b in terms of d. Let me write a = d * m and b = d * n, where m and n are integers that are coprime, meaning their GCD is 1. That makes sense because if m and n had a common divisor, then d wouldn't be the greatest common divisor of a and b.Now, since a and b are expressed as d*m and d*n, their difference is given as 120. So, |a - b| = 120. Without loss of generality, let's assume that a > b, so a - b = 120. Therefore, d*m - d*n = 120, which simplifies to d*(m - n) = 120. So, d is a divisor of 120.Also, since LCM(a, b) = 105d, and LCM(a, b) is equal to (a*b)/GCD(a, b). Substituting the values, LCM(a, b) = (d*m * d*n)/d = d*m*n. So, d*m*n = 105d. Dividing both sides by d, we get m*n = 105.So now, I have two equations:1. d*(m - n) = 1202. m*n = 105And m and n are coprime positive integers with m > n.So, my task now is to find integers m and n such that their product is 105 and their difference (m - n) is 120/d. Since m and n are coprime, I need to find pairs of factors of 105 that are coprime.Let me list the positive integer pairs (m, n) such that m*n = 105 and m > n. The factors of 105 are 1, 3, 5, 7, 15, 21, 35, 105.So, possible pairs are:1. (105, 1)2. (21, 5)3. (15, 7)4. (35, 3)Wait, let me check:105 can be factored as:1 * 1053 * 355 * 217 * 15So, the coprime pairs are:(105, 1): GCD is 1.(35, 3): GCD is 1.(21, 5): GCD is 1.(15, 7): GCD is 1.So, all these pairs are coprime. So, that's four possible pairs.Now, for each pair, I can compute m - n and then find d = 120 / (m - n). Since d must be a positive integer, (m - n) must divide 120.Let me compute m - n for each pair:1. (105, 1): m - n = 1042. (35, 3): m - n = 323. (21, 5): m - n = 164. (15, 7): m - n = 8So, m - n is 104, 32, 16, or 8.Then, d = 120 / (m - n). Let's compute d for each case:1. For m - n = 104: d = 120 / 104 = 120 ÷ 104. Let me compute that. 104 goes into 120 once, with a remainder of 16. So, 120/104 simplifies to 15/13. Hmm, that's not an integer. So, this case is invalid because d must be an integer.2. For m - n = 32: d = 120 / 32 = 120 ÷ 32. 32 goes into 120 three times (32*3=96), remainder 24. So, 120/32 = 15/4. Again, not an integer. So, invalid.3. For m - n = 16: d = 120 / 16 = 7.5. Hmm, that's 7.5, which is not an integer. So, invalid.4. For m - n = 8: d = 120 / 8 = 15. That's an integer. So, this is a valid case.So, the only valid case is when m - n = 8, which gives d = 15.So, now, with d = 15, and m*n = 105, and m - n = 8.So, let me find m and n.We have:m + n = ?Wait, actually, we have m - n = 8 and m*n = 105.We can set up equations:Let me denote m = n + 8.Then, substituting into m*n = 105:(n + 8)*n = 105n^2 + 8n - 105 = 0Now, solving this quadratic equation:n = [-8 ± sqrt(64 + 420)] / 2Because discriminant D = 64 + 420 = 484sqrt(484) = 22So, n = [-8 + 22]/2 = 14/2 = 7Or, n = [-8 - 22]/2 = -30/2 = -15But since n is positive, n = 7.Therefore, m = n + 8 = 7 + 8 = 15.So, m = 15, n = 7.So, now, a = d*m = 15*15 = 225b = d*n = 15*7 = 105Wait, let me check:a = 225, b = 105Difference: 225 - 105 = 120. Correct.GCD of 225 and 105: Let's compute.Prime factors of 225: 15*15 = 3^2 * 5^2Prime factors of 105: 3*5*7So, GCD is the minimum exponents: 3^1 * 5^1 = 15. Correct.LCM of 225 and 105: The maximum exponents: 3^2 * 5^2 * 7^1 = 9*25*7 = 1575Now, 105 times GCD: 105 * 15 = 1575. Correct.So, that's consistent.Therefore, the larger number is 225.Wait, but just to make sure, let me check if there are any other possible pairs.Earlier, when m - n was 104, 32, 16, only 8 gave integer d. So, that's the only solution.Therefore, the larger number is 225.**Final Answer**The larger number is boxed{225}.

answer：First, I recognize that when each of the four positive integers is divided by 11, they leave remainders of 2, 4, 6, and 8 respectively.To find the remainder when the sum of these four integers is divided by 11, I can add the remainders together first.Adding the remainders: 2 + 4 + 6 + 8 equals 20.Next, I divide this sum of remainders by 11 to determine the final remainder. When 20 is divided by 11, the quotient is 1 and the remainder is 9.Therefore, the remainder when the sum of the four integers is divided by 11 is 9.

answer：Alright, so I have this problem where I need to find the value of γ - α given that α, β, γ are angles between 0 and 2π, and for any real number x, the sum of cos(x + α) + cos(x + β) + cos(x + γ) equals zero. Hmm, okay. Let me try to break this down step by step.First, the equation given is:cos(x + α) + cos(x + β) + cos(x + γ) = 0 for all x ∈ ℝ.Since this holds for any real x, it must be true regardless of the value of x. That suggests that the expression is identically zero, meaning the coefficients of the cosine and sine terms must each be zero. Maybe I can use some trigonometric identities to simplify this expression.I remember that cos(A + B) can be expanded as cos A cos B - sin A sin B. So, let me apply that to each term:cos(x + α) = cos x cos α - sin x sin αSimilarly,cos(x + β) = cos x cos β - sin x sin βcos(x + γ) = cos x cos γ - sin x sin γSo, adding all three together:[cos x cos α - sin x sin α] + [cos x cos β - sin x sin β] + [cos x cos γ - sin x sin γ] = 0Let me factor out cos x and sin x:cos x (cos α + cos β + cos γ) - sin x (sin α + sin β + sin γ) = 0Since this equation must hold for all x, the coefficients of cos x and sin x must each be zero. Otherwise, the equation would not hold for all x. So, we can set up the following system of equations:1. cos α + cos β + cos γ = 02. sin α + sin β + sin γ = 0So, both the sum of the cosines and the sum of the sines of these angles must be zero.Hmm, okay. So, we have two equations:cos α + cos β + cos γ = 0sin α + sin β + sin γ = 0I need to find γ - α. Maybe I can relate these equations somehow.I recall that if the sum of three unit vectors is zero, they form an equilateral triangle on the unit circle. Is that applicable here? Because each cosine and sine term can be thought of as the x and y components of a unit vector at angle α, β, γ respectively.So, if we have three vectors:(cos α, sin α), (cos β, sin β), (cos γ, sin γ)And their sum is (0, 0), then these three vectors must form a closed triangle, meaning they are the vertices of an equilateral triangle on the unit circle.Wait, but in that case, the angles between each pair of vectors would be 120 degrees apart, or 2π/3 radians. So, the difference between each angle would be 2π/3.But in our case, we have three angles α, β, γ with 0 < α < β < γ < 2π. So, if they are equally spaced around the circle, each separated by 2π/3, then we can write:β = α + 2π/3γ = α + 4π/3But let's check if this makes sense.If β = α + 2π/3 and γ = α + 4π/3, then the differences between consecutive angles are 2π/3 each. So, the total around the circle would be 2π, which is correct because 3*(2π/3) = 2π.But in the problem statement, it's given that α < β < γ, so γ would be the largest angle. But if γ = α + 4π/3, then depending on α, γ could potentially exceed 2π. Wait, but the problem states that γ < 2π, so α must be less than 2π - 4π/3 = 2π/3. So, α must be less than 2π/3.Alternatively, maybe the angles are not necessarily equally spaced, but just arranged such that their vector sum is zero. So, perhaps they form a triangle, but not necessarily equilateral.Wait, but for three vectors to sum to zero, they must form a triangle, but the triangle doesn't have to be equilateral. However, in this case, since all three vectors are of unit length, the only way their sum is zero is if they form an equilateral triangle. Because if they don't, the vectors won't cancel each other out.Wait, is that necessarily true? Let me think.Suppose we have three unit vectors with angles α, β, γ. If their sum is zero, then they must form a triangle when connected tip-to-tail. For three vectors of equal length, the only way their sum is zero is if they are symmetrically placed, i.e., each 120 degrees apart. So, yes, I think that's correct. So, the angles must be spaced 2π/3 apart.Therefore, the angles α, β, γ must satisfy:β = α + 2π/3γ = α + 4π/3But let's verify this.If β = α + 2π/3 and γ = α + 4π/3, then let's compute cos α + cos β + cos γ.cos α + cos(α + 2π/3) + cos(α + 4π/3)Similarly, sin α + sin(α + 2π/3) + sin(α + 4π/3)I can use the identity for sum of cosines with equally spaced angles.I remember that the sum of cos(θ + kφ) for k=0,1,2 is zero if φ = 2π/3. Similarly for sine.Yes, so:cos α + cos(α + 2π/3) + cos(α + 4π/3) = 0Similarly,sin α + sin(α + 2π/3) + sin(α + 4π/3) = 0So, that satisfies both equations. Therefore, this must be the case.Therefore, the angles are equally spaced around the circle, each 2π/3 apart.So, β = α + 2π/3, γ = α + 4π/3.Therefore, γ - α = 4π/3.Wait, but let me check if γ is less than 2π.Since γ = α + 4π/3, and α > 0, so γ must be less than 2π.Therefore, α + 4π/3 < 2π => α < 2π - 4π/3 = 2π/3.So, α must be less than 2π/3, which is consistent with the given condition 0 < α < β < γ < 2π.So, yes, this seems to hold.Therefore, γ - α = 4π/3.Wait, but let me think again. Is there another possibility where the angles are not equally spaced but still satisfy the sum being zero?Suppose two angles are the same, but then the third would have to be such that it cancels out the sum. But since α < β < γ, all three angles are distinct. So, they can't be the same.Alternatively, maybe two angles are symmetric with respect to some axis, but the third is placed such that the sum cancels. But for three vectors, the only way their sum is zero is if they are equally spaced, as otherwise, their vector sum won't cancel out.Therefore, I think the conclusion is that γ - α = 4π/3.But let me double-check.Suppose I set α = 0, then β = 2π/3, γ = 4π/3.Then cos(0) + cos(2π/3) + cos(4π/3) = 1 + (-1/2) + (-1/2) = 0Similarly, sin(0) + sin(2π/3) + sin(4π/3) = 0 + (√3/2) + (-√3/2) = 0So, that works.Alternatively, if α is not zero, say α = π/6, then β = π/6 + 2π/3 = 5π/6, γ = π/6 + 4π/3 = 3π/2.Then cos(π/6) + cos(5π/6) + cos(3π/2) = (√3/2) + (-√3/2) + 0 = 0Similarly, sin(π/6) + sin(5π/6) + sin(3π/2) = (1/2) + (1/2) + (-1) = 0So, that also works.Therefore, regardless of α, as long as the angles are spaced 2π/3 apart, the sum is zero.Hence, γ - α = 4π/3.Wait, but 4π/3 is more than π. Is that acceptable? Since γ is less than 2π, and α is positive, yes, 4π/3 is less than 2π.Wait, 4π/3 is approximately 4.188 radians, which is less than 2π (approximately 6.283 radians). So, yes, that's acceptable.Therefore, the value of γ - α is 4π/3.**Final Answer**The value of (gamma - alpha) is (boxed{dfrac{4pi}{3}}).

answer：Okay, so I need to calculate the half-life, but the problem gives me an equation PV = vRT and some values. Hmm, let me see. First, I should probably understand what each of these variables represents. PV is pressure multiplied by volume, v is specific volume, R is the gas constant, and T is temperature. Wait, actually, in the ideal gas law, it's usually PV = nRT, where n is the number of moles. But here, it's written as PV = vRT. Maybe v here represents the specific volume, which is volume per unit mass? Or perhaps it's a typo and should be nRT? Hmm, the given pressure calculation seems to use moles, so maybe v is the specific volume. Let me check the units.The pressure is calculated as P = 30 moles * 8.31 J/mol·K * 300 K / 0.06648 m³. Let me compute that step by step. First, 30 moles times 8.31 J/mol·K is 30 * 8.31 = 249.3 J/K. Then, multiplying by 300 K gives 249.3 * 300 = 74,790 J. Dividing that by 0.06648 m³ gives 74,790 / 0.06648 ≈ 1,125,000 Pa. So, P is 1,125,000 Pa, which is written as 1125 kPa. Okay, so that makes sense. Now, the question is about calculating the half-life. But I don't see any information about decay or half-life in the given equations. Maybe this is part of a larger problem where the half-life is related to the pressure or something else? Wait, perhaps this is about the half-life of a gas escaping from a container or something like that? If so, maybe it's related to the pressure decreasing over time, and we need to find the time it takes for the pressure to drop by half. If that's the case, then maybe we can model the pressure decay as an exponential decay process. The general formula for exponential decay is P(t) = P0 * e^(-kt), where P0 is the initial pressure, k is the decay constant, and t is time. The half-life (t₁/₂) is the time it takes for P(t) to reach P0/2. So, setting P(t₁/₂) = P0/2, we have:P0/2 = P0 * e^(-k t₁/₂)Divide both sides by P0:1/2 = e^(-k t₁/₂)Take the natural logarithm of both sides:ln(1/2) = -k t₁/₂So,t₁/₂ = ln(2) / kBut I don't have the decay constant k here. Maybe I need to find k from some other information? The problem only gives me the initial pressure calculation. Alternatively, maybe the half-life is related to the time it takes for the volume to change, but without more information about how the volume or pressure changes over time, I can't directly compute the half-life. Wait, perhaps the problem is about the half-life of a reaction, and PV = vRT is part of the reaction conditions. If so, maybe we need to find the half-life using the rate law, but again, without information about the reaction rate or order, I can't proceed. Hmm, maybe I'm overcomplicating this. The problem gives PV = vRT and calculates P as 1125 kPa. Maybe the half-life is related to the time it takes for the pressure to drop to half of 1125 kPa, but without knowing the rate at which the pressure decreases, I can't find the half-life. Alternatively, perhaps this is a problem about radioactive decay, and the pressure is somehow related to the number of moles decaying. If the gas is produced by a radioactive decay process, then the number of moles would decrease over time, affecting the pressure. In that case, the number of moles n(t) would follow n(t) = n0 * e^(-λ t), where λ is the decay constant. Since PV = nRT, the pressure P(t) would be proportional to n(t), so P(t) = P0 * e^(-λ t). Then, the half-life would be t₁/₂ = ln(2) / λ. But again, without knowing λ or the rate of decay, I can't calculate the half-life. Wait, maybe the problem is missing some information? It just gives PV = vRT and calculates P as 1125 kPa. Maybe the half-life is for the pressure to decrease by half due to some process, but without knowing the process or the rate, I can't determine the half-life. Alternatively, perhaps the half-life is for the volume to change, but again, without more context, it's unclear. Wait, maybe the problem is about the half-life of a gas in a container with a leak. If the leak follows first-order kinetics, then the pressure would decrease exponentially. But without knowing the leak rate or the time constant, I can't compute the half-life. Alternatively, maybe the half-life is for the temperature to change, but that doesn't make much sense in this context. Hmm, I'm stuck. The problem gives me an initial pressure calculation but doesn't provide any information about how the system changes over time or any decay process. Without additional data, I can't calculate the half-life. Maybe I need to assume that the half-life is related to the time it takes for the pressure to drop by half, but without knowing the rate or the process, I can't find the numerical value. Alternatively, perhaps the half-life is for the reaction itself, and the pressure is just a given condition. If that's the case, and if it's a first-order reaction, then the half-life is independent of the initial concentration (or pressure). But without knowing the rate constant, I still can't compute it. Wait, maybe the problem is part of a larger context where the half-life was previously discussed, and this is just a step in the calculation? But since I only have this information, I can't proceed. I think I need to ask for clarification or more information. Without knowing the process or the rate constant, I can't calculate the half-life. But since the problem specifically asks to calculate the half-life given PV = vRT and the pressure calculation, maybe I'm missing something. Perhaps the half-life is related to the time it takes for the volume to change by half, but again, without knowing the rate of volume change, I can't determine it. Alternatively, maybe the half-life is for the temperature to change, but that seems unrelated. Wait, perhaps the problem is about the half-life of a gas being absorbed or something, but without knowing the absorption rate, I can't compute it. I'm going in circles here. Maybe I should look back at the given information and see if I can interpret it differently. The given equation is PV = vRT. If v is specific volume, then v = V/m, where m is mass. But in the calculation, they used moles, so maybe v is molar volume? Or perhaps it's a typo and should be nRT. Wait, in the calculation, they used n = 30 moles, R = 8.31 J/mol·K, T = 300 K, and V = 0.06648 m³. So, P = nRT / V. That makes sense. So, maybe the equation should be PV = nRT, not vRT. So, perhaps it's a typo, and the correct equation is PV = nRT. If that's the case, then the given pressure is 1125 kPa. But still, how does that relate to the half-life? Maybe the half-life is for the number of moles to decrease by half, which would cause the pressure to drop by half as well, assuming temperature and volume are constant. If the number of moles decreases due to a decay process, then yes, the pressure would decrease proportionally. So, if n(t) = n0 * e^(-λ t), then P(t) = P0 * e^(-λ t). The half-life would be t₁/₂ = ln(2) / λ. But again, without knowing λ, I can't compute the half-life. Alternatively, if the process is second-order or zero-order, the half-life formula would be different. But without knowing the order of the reaction or the rate constant, I can't proceed. Wait, maybe the problem is about the half-life of a gas in a container with a fixed volume and temperature, and the pressure is given. If the gas is leaking out, perhaps the leak rate is proportional to the pressure, leading to an exponential decay. In that case, the pressure would decrease as P(t) = P0 * e^(-kt), and the half-life would be t₁/₂ = ln(2) / k. But without knowing k, which depends on the leak rate, I can't find the half-life. Alternatively, if the leak rate is given by some other parameter, like the flow rate, I could find k. But since no such information is provided, I can't proceed. Hmm, maybe I need to make an assumption. If the half-life is for the pressure to drop by half, and if the process is exponential decay, then t₁/₂ = ln(2) / k. But without knowing k, I can't compute it. Alternatively, if the problem is about the half-life of a reaction where the pressure is a function of the extent of reaction, then maybe I can relate the pressure change to the reaction progress. But without the reaction equation or rate law, I can't do that. Wait, maybe the problem is about the half-life of a gas being consumed in a reaction, and the pressure is related to the amount of gas. If it's a first-order reaction, then the half-life is independent of the initial pressure. But again, without the rate constant, I can't compute it. I think I'm stuck because the problem doesn't provide enough information. It gives me the initial pressure calculation but doesn't specify the process or the rate at which the pressure changes. Without that, I can't determine the half-life. Maybe I should look for similar problems or standard formulas. For example, in gas leaks, the time to reach a certain pressure can be modeled, but without knowing the leak rate or the volume, it's impossible. Alternatively, if this is about radioactive decay of a gas, and the pressure is due to the decay, then the half-life would be related to the decay constant. But again, without knowing the decay constant or the initial and final pressures over time, I can't compute it. Wait, perhaps the problem is simpler. Maybe the half-life is for the pressure to drop by half, and the rate is given implicitly by the ideal gas law. But I don't see how. Alternatively, maybe the half-life is for the volume to change, but without knowing how the volume changes over time, I can't compute it. I think I need to conclude that the problem as given doesn't provide enough information to calculate the half-life. There's no information about the process causing the change (decay, reaction, leak, etc.), the rate of change, or any time-dependent data. Therefore, I can't calculate the half-life with the given information. Maybe I missed something, but based on what's provided, it's insufficient.