Final answer:
The approximate time is 0.88 seconds. To find the time it will take to reach the maximum height of the toy rocket, we can calculate the derivative of the height equation. The exact time is 1.5√2 seconds, which approximates to 3.36 seconds. By plugging in this time, we can find the maximum height of the rocket, which is approximately 20.89 feet. To find the time it takes for the rocket to strike the ground, we need to solve the equation when the height is zero.
Step-by-step explanation:
To find the time it will take to reach the maximum height, we need to compute the time derivative of the rocket's height function. So let's take the derivative of the equation s = -16t^2 + 48√2t:
s' = -32t + 48√2
Next, we set the derivative equal to zero to find the time at which the rocket reaches its maximum height: -32t + 48√2 = 0
Solving for t, we find t = 1.5√2 seconds. This is the exact answer to part a. To approximate this answer to two decimal places, we can substitute in the value of √2 (approximately 1.41) and find that t ≈ 3.36 seconds.
The maximum height of the rocket can be found by substituting the value of t into the original equation: s = -16(1.5√2)^2 + 48√2(1.5√2). Simplifying, we get s ≈ 20.89 feet.
To find the time it will take for the rocket to strike the ground, we need to find the value of t when s = 0. Using the quadratic formula, we get two solutions: t ≈ 0.88 seconds and t ≈ 2.12 seconds. Rounding the answer to two decimal places, the rocket will take approximately 0.88 seconds to strike the ground.