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📅 July 6, 2026 👤 scraface

Jul 06, 2026

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📌 Since $72 \mid 432$ (because $432 \div 72 = 6$), the GCD is $72$.

📌 Alternatively, $432 = 6 \cdot 72$, so $\gcd = 72$.

📌 Thus, the answer is $\boxed{72}$.Question: A renewable energy startup is designing a rectangular solar panel array with a fixed perimeter of 80 meters. What is the maximum possible area of the array?

📌 Solution: The perimeter of a rectangle is given by $ P = 2(l + w) $, where $ l $ is length and $ w $ is width. Given $ 2(l + w) = 80 $, we simplify to $ l + w = 40 $. The area $ A = lw $. Express $ w = 40 - l $, so $ A = l(40 - l) = 40l - l^2 $. This quadratic equation opens downward, and its maximum occurs at the vertex $ l = \frac{-b}{2a} = \frac{-40}{-2} = 20 $. Substituting $ l = 20 $, $ w = 20 $, so $ A = 20 \times 20 = 400 $. The maximum area is $\boxed{400}$.

📌 Question: An entomologist studying pollination patterns models the number of flowers visited by bees as $ f(t) = -5t^2 + 30t + 100 $. What is the maximum number of flowers the bees visit in a day?

📌 Solution: The function $ f(t) = -5t^2 + 30t + 100 $ is a quadratic opening downward. The vertex occurs at $ t = \frac{-b}{2a} = \frac{-30}{2(-5)} = 3 $. Substituting $ t = 3 $, $ f(3) = -5(9) + 30(3) + 100 = -45 + 90 + 100 = 145 $. The maximum number of flowers visited is $\boxed{145}$.

📌 Question: A robotics engineer designs a robot whose path follows the hyperbola $ 9x^2 - 36x - 4y^2 + 24y = 36 $. Find the center of the hyperbola.

📌 Solution: Group and complete the square for $ x $ and $ y $.

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