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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 $.
For $ x $: $ 9(x^2 - 4x) = 9[(x - 2)^2 - 4] $.
For $ y $: $ -4(y^2 - 6y) = -4[(y - 3)^2 - 9] $.
Substitute back: $ 9(x - 2)^2 - 36 - 4(y - 3)^2 + 36 = 36 $. Simplify: $ 9(x (x - 2)^2 - 4) - 4(y - 3)^2 = 36 $. Divide by 36: $ \frac{(x - 2)^2}{4} - \frac{(y - 3)^2}{9} = 1 $. The center is at $ (2, 3) $. The center is $\boxed{(2, 3)}$.
Question: For all real numbers $ x $, $ y $, and $ z $, find the number of functions $ f: \mathbb{R} \to \mathbb{R} $ satisfying $ f(x + y) + f(z) = f(x + z) + f(y) $.