Rachael shoots a rocket straight upward with an initial velocity of 4 meters per second from a height of 180 meters. The distance d (in meters) of the ball from the ground after t seconds is given by the equation d = 180 + 4t - 0.5g(t²), where gravity g is 9.8 m/s². After approximately how many seconds does the rocket strike the ground?(A) 5 seconds(B) 6.5 seconds(C) 8.5 seconds(D) 9 seconds(E) 10 seconds
Question
Rachael shoots a rocket straight upward with an initial velocity of 4 meters per second from a height of 180 meters. The distance d (in meters) of the ball from the ground after t seconds is given by the equation d = 180 + 4t - 0.5g(t²), where gravity g is 9.8 m/s². After approximately how many seconds does the rocket strike the ground?(A) 5 seconds(B) 6.5 seconds(C) 8.5 seconds(D) 9 seconds(E) 10 seconds
Solution
To find out when the rocket strikes the ground, we need to set the distance d to 0 (since the rocket is on the ground) and solve for t.
The equation becomes:
0 = 180 + 4t - 0.59.8t² 0 = 180 + 4t - 4.9t²
Rearranging the equation gives:
4.9t² - 4t - 180 = 0
This is a quadratic equation in the form of at² - bt - c = 0, where a = 4.9, b = -4, and c = -180.
We can solve this equation using the quadratic formula:
t = [-b ± sqrt(b² - 4ac)] / (2a)
Substituting the values gives:
t = [4 ± sqrt((-4)² - 44.9(-180))] / (2*4.9) t = [4 ± sqrt(16 + 3528)] / 9.8 t = [4 ± sqrt(3544)] / 9.8 t = [4 ± 59.53] / 9.8
This gives two possible solutions:
t = 63.53 / 9.8 ≈ 6.48 seconds t = -55.53 / 9.8 ≈ -5.67 seconds
Since time cannot be negative, the rocket strikes the ground after approximately 6.5 seconds. So, the answer is (B) 6.5 seconds.
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