Question

Every year in delaware there is a contest where people create cannons and catapults designed to launch pumpkins as far in the air as possible. the equation y=10+95x-16x^2 can be used to represent the height,y, of a launched pumpkin, where x is the time in seconds that the pumpkin has been in the air. What is the maximum height that the pumpkin reaches? How many seconds have passed when the pumpkin hits the ground?
1.The pumpkin's maximum height is 2.97 feet and it hits the ground after 6.04 seconds
2.The pumpkin's maximum height is 151.02 feet and it hits the ground after 6.04 seconds
3.The pumpkin's maximum height is 2.97 feet and it hits the ground after 151.02 seconds
4.The pumpkin's maximum height is 151.02 feet and it hits the ground after 2.97 seconds

Answer 1

For this case we have the following function that models the problem is:
 $y = 10 + 95x-16x ^ 2$
 The time for which the maximum height occurs is obtained by deriving the equation:
 $y '= 95-32x$
 We equal zero and clear x:
 $95-32x = 0 32x = 95 x = 95/32 x = 2.97$
 The maximum height is obtained by evaluating the time obtained in the given equation:
 $y = 10 + 95 * (2.97) -16 * (2.97) ^ 2 y = 151.02$
 Then, by the time it reaches the ground we have:
 $10 + 95x-16x ^ 2 = 0$
 From here, we clear x solving the quadratic equation:
 $x = \frac{-95+/- \sqrt{95^2-4(-16)(10)} }{2(-16)}$
 We discard the negative root, and obtain as a result:
 $x = 6.04$
 Answer:
 2.The pumpkin's maximum height is 151.02 feet and it hits the ground after 6.04 seconds