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Rachael shoots a rocket straight upward with an initial velocity of 8 meters per second from a height of 20 meters. The distance d (in meters) of the ball from the ground after t seconds is given by the equation d = 20 + 8t - 0.5g(t²), where gravity g is 9.8 m/s². After approximately how many seconds does the rocket strike the ground?(A)  0.5 seconds(B) 1 second(C) 1.5 seconds(D) 2 seconds(E) 3 seconds

Question

Rachael shoots a rocket straight upward with an initial velocity of 8 meters per second from a height of 20 meters. The distance d (in meters) of the ball from the ground after t seconds is given by the equation d = 20 + 8t - 0.5g(t²), where gravity g is 9.8 m/s². After approximately how many seconds does the rocket strike the ground?(A)  0.5 seconds(B) 1 second(C) 1.5 seconds(D) 2 seconds(E) 3 seconds

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Solution

The rocket strikes the ground when the distance d is equal to 0. So, we need to solve the equation for t when d = 0.

0 = 20 + 8t - 0.59.8t² 0 = 20 + 8t - 4.9t²

Rearranging the terms, we get:

4.9t² - 8t - 20 = 0

This is a quadratic equation in the form of at² - bt - c = 0. We can solve for t using the quadratic formula:

t = [b ± sqrt(b² - 4ac)] / 2a

Substituting the values a = 4.9, b = -8, and c = -20 into the formula, we get:

t = [8 ± sqrt((-8)² - 44.9(-20))] / 2*4.9 t = [8 ± sqrt(64 + 392)] / 9.8 t = [8 ± sqrt(456)] / 9.8 t = [8 ± 21.35] / 9.8

We get two solutions: t = 2.99 seconds and t = -1.36 seconds. Since time cannot be negative, we discard the negative solution.

So, the rocket strikes the ground after approximately 3 seconds. Therefore, the answer is (E) 3 seconds.

This problem has been solved

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