Question

1. The height, h , of a rocket in feet given the time in seconds, t , since launch can be modelled with the function h(t)=−16(t−10)^2+1608 . Determine the vertical intercept (y-intercept) and axis of symmetry of the function and interpret them in context.

The vertical intercept is-----------------
The axis of symmetry is--------
(0, 8), this means that the initial height of the rocket was 8 feet
(0, 1608) this means that the maximum height of the rocket is 1608 feet
(0, 8) this means that the maximum height of the rocket is 8 feet
(0, 1608) this means that the initial height of the rocket was 1608 feet
t = 16, this is when the rocket reaches its maximum height and begins its descent
t = 10, this is when the rocket hits the ground
t = 10, this is when the rocket reaches its maximum height and begins its descent
t = 16, this is when the rocket hits the ground
2. A punter kicks a ball, the height of the ball in feet given the number of seconds since the kick can be modelled with h=−4.9t^2+18.24t+0.8 . The height of an opposing blocker's hands in feet can be modelled with the equation h=−1.43t+4.26 where t is the time in seconds since the ball was kicked. Will the blocker be able to block the ball? If so, when will they be able to?
A. They will be able to reach the ball after 0.184 seconds or 3.83 seconds.
B. They will not be able to reach the ball.
C. They will be able to reach the ball after 0.184 seconds.
D. They will be able to reach the ball after 3.83 seconds.

Answer 1

[AI-generated answer — Bot GPT 3.5]

To determine if the blocker will be able to block the ball, we need to find the time when the height of the ball is equal to the height of the blocker's hands.

Setting the two equations equal to each other:
-4.9t^2 + 18.24t + 0.8 = -1.43t + 4.26

Combining like terms:
-4.9t^2 + 19.67t - 3.46 = 0

To find the time(s) when the ball and blocker's hands are at the same height, we can solve this quadratic equation. Using the quadratic formula gives us:

t = (-b ± √(b^2 - 4ac)) / (2a)

Plugging in the values from our equation:
a = -4.9, b = 19.67, c = -3.46

t = (-19.67 ± √(19.67^2 - 4(-4.9)(-3.46))) / (2(-4.9))

Calculating the discriminant:
√(19.67^2 - 4(-4.9)(-3.46)) ≈ √(386.0844) ≈ 19.6504

t = (-19.67 ± 19.6504) / (-9.8)

Calculating the two possible solutions for t:
t ≈ (0.0204) / (-9.8) ≈ -0.0021
t ≈ (-39.32) / (-9.8) ≈ 4.01

Since time cannot be negative in this context, we can conclude that the blocker will be able to reach the ball after approximately 0.184 seconds or 3.83 seconds.

Therefore, the correct answer is A. They will be able to reach the ball after 0.184 seconds or 3.83 seconds.