Final answer:
To determine how many seconds it takes for the firework shell to first reach a height of 300 meters, we need to solve the given function. The quadratic formula can help us find the time when the shell reaches the desired height. After calculating, we find that it takes approximately 5.2 seconds for the shell to reach a height of 300 meters.
Step-by-step explanation:
To determine how many seconds it takes for the firework shell to first reach a height of 300 meters, we need to solve the equation f(t) = -4.9t² + 80t = 300. Rearranging the equation, we have -4.9t² + 80t - 300 = 0. This is a quadratic equation, so we can solve it by factoring, completing the square, or using the quadratic formula. Once we solve the equation, we will find the value of t, which represents the time in seconds when the shell first reaches a height of 300 meters.
For example, let's use the quadratic formula to solve the equation. The quadratic formula is t = (-b ± √(b² - 4ac)) / (2a), where a = -4.9, b = 80, and c = -300. Plugging in these values, we get t ≈ 5.2 seconds or t ≈ 12.8 seconds.
Since we are looking for the first time the shell reaches a height of 300 meters, we take the smaller positive value, t ≈ 5.2 seconds. Therefore, it takes approximately 5.2 seconds for the shell to reach a height of 300 meters.