The first thing “e” wanted to do is to “Calculuize” his favourite game, Super Mario. So, one day, his friend 𝑝𝑖 worked with e to do the following experiment. They stood 3 m apart and they were also 1.5 m vertically from the ground when e decided to throw a ball to his friend 𝑝𝑖. The path of the ball formed a parabola of the form 𝑦 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐, where x represents the horizontal distance the ball travelled and y represents the height above the ground. “e” being the genius was able to determine when 𝑥 = 1 𝑡ℎ𝑒 𝑠𝑙𝑜𝑝𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑡𝑎𝑛𝑔𝑒𝑛𝑡 𝑡𝑜 𝑡ℎ𝑒 𝑝𝑎𝑡ℎ 𝑜𝑓 𝑡ℎ𝑒 𝑏𝑎𝑙𝑙 𝑤𝑎𝑠 1. Using this information and using calculus methods only, determine the maximum height of the ball. Show all your work.
y = ax^2 + bx + c
since y(0) = 1.5,
y = ax^2 + bx + 3/2
y' = 2ax + b
at x=1, y'=1, so
2a+b = 1
so y = ax^2 + (1-2a)x + 3/2
since the parabola is symmetric about the line x = 3/2,
-b/2a = 3/2
-b/(1-b) = 3/2
b = 3
a = -1
y = -x^2 + 3x + 3/2
check: y'(1) = -2+3 = 1