Question

Every year in Delaware, there is a contest where people create cannons and catapults designed to launch pumpkins as far into the air as possible. The equation y = 12 + 105x - 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?

a) The maximum height is 125 feet, and it hits the ground after 7 seconds.
b) The maximum height is 125 feet, and it hits the ground after 6 seconds.
c) The maximum height is 145 feet, and it hits the ground after 7 seconds.
d) The maximum height is 145 feet, and it hits the ground after 6 seconds.

Answer 1

The maximum height of the pumpkin is 125 feet and it hits the ground after 7 seconds.

Step-by-step explanation:

The equation given is y = 12 + 105x - 16x^2, where y represents the height of a launched pumpkin and x represents the time in seconds. To find the maximum height, we need to determine the vertex of the parabolic equation. The vertex can be found using the formula x = -b/2a, where a, b, and c are the coefficients of the quadratic equation. Plugging in the values, we get x = -105/(2*(-16)) = 6.5625 seconds. Substituting this value back into the equation, we can find the maximum height y = 12 + 105*6.5625 - 16*6.5625^2 = 125 feet. The pumpkin hits the ground when y = 0, so we solve the equation 12 + 105x - 16x^2 = 0 for x. Solving this quadratic equation, we get x = 7 seconds. Therefore, the correct answer is (a) The maximum height is 125 feet, and it hits the ground after 7 seconds.