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Home » Calculate \( 1.12^8 \approx 2.4760 \) - Coaching Toolbox

Calculate \( 1.12^8 \approx 2.4760 \) - Coaching Toolbox

📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 Solution: Divide numerator and denominator by $ n^2 $: $ g(n) = \frac{3 + \frac{5}{n}}{1 + \frac{1}{n^2}} $. As $ n \to \infty $, $ \frac{5}{n} \to 0 $ and $ \frac{1}{n^2} \to 0 $. Thus, the limit is $ \frac{3}{1} = \boxed{3} $.

📌 Question: A civil engineer designs a triangular support beam with side lengths satisfying $ a + b = 10 $ and $ a^2 + b^2 = 58 $. Find the area of the triangle.

📌 Solution: Let $ a + b = 10 $ and $ a^2 + b^2 = 58 $. Compute $ (a + b)^2 = 100 = a^2 + 2ab + b^2 = 58 + 2ab $. Solve for $ ab $: $ 2ab = 42 \Rightarrow ab = 21 $. The area is $ \frac{1}{2}ab = \boxed{\frac{21}{2}} $.

📌 Question: A data scientist optimizes a recommendation algorithm with the inequality $ 2x^2 - 7x + 3 < 0 $. Solve for $ x $.

📌 Solution: Factor $ 2x^2 - 7x + 3 $. Find roots: $ x = \frac{7 \pm \sqrt{49 - 24}}{4} = \frac{7 \pm 5}{4} \Rightarrow x = 3 $ or $ x = \frac{1}{2} $. The quadratic opens upwards, so the inequality holds between the roots: $ \boxed{\left( \frac{1}{2}, 3 \right)} $.

📌 Question: A bioinformatician studies a DNA sequence where the probability of a mutation at position $ n $ is $ p_n = \frac{n}{n^3 + 2n} $. Simplify $ p_n $ and find its limit as $ n \to \infty $.

📌 Solution: Simplify $ p_n = \frac{n}{n(n^2 + 2)} = \frac{1}{n^2 + 2} $. As $ n \to \infty $, $ n^2 + 2 \to \infty $, so $ \lim_{n \to \infty} p_n = \boxed{0} $.

📌 Question: A civil engineer calculates the stress on a beam with cross-section $ A = \frac{4}{x} - x^2 $, where $ x > 0 $. Find the maximum stress.

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