You launch a firework from the roof of a 50 foot building that goes up 30 feet and has an initial velocity of 5.4 ft./s how long after setting off the firework should the delay be set. at what time will the fireworks explode based on the height you wanted it to explode consider where that height fits into the equation and solve for time. you must show your work.
To determine how long after setting off the firework the delay should be set, we need to find the time it takes for the firework to reach the desired height before exploding.
The height of the firework above the ground at any time can be modeled by the equation: h(t) = -16t^2 + v0t + h0, where t is the time in seconds, v0 is the initial velocity in ft./s, and h0 is the initial height in feet.
Given that the firework is launched from the roof of a 50-foot building, the initial height is 50 feet (h0 = 50). The initial velocity is 5.4 ft./s (v0 = 5.4).
Let's denote the time it takes for the firework to reach the desired exploding height as t_explosion.
We want to solve for t_explosion when the height is 30 feet.
Plug in the values into the equation: 30 = -16t_explosion^2 + 5.4t_explosion + 50
By rearranging the equation and setting it equal to zero, we get: -16t_explosion^2 + 5.4t_explosion + 20 = 0
To solve this quadratic equation, we can either factor it or use the quadratic formula. In this case, factoring is not straightforward, so let's use the quadratic formula: t_explosion = (-b ± √(b^2 - 4ac)) / 2a
For our equation, a = -16, b = 5.4, and c = 20. Plugging in these values, we get: t_explosion = (-5.4 ± √(5.4^2 - 4(-16)(20))) / (2(-16))
Calculating this equation will give us two solutions, but we are only interested in the positive time value since time cannot be negative in this context.
t_explosion ≈ 2.67 seconds
Therefore, the delay should be set to approximately 2.67 seconds after setting off the firework for it to explode at the desired height.