If a toy rocket is launched vertically upward from ground level with an initial velocity of 148 feet per second, then its height h after t seconds is given by the equation h(t)=-16t2+148t. How long will it take the rocket to return to the ground?

To find the time it takes for the rocket to return to the ground, we need to find the value of t when the height h equals zero. We can do this by setting the equation h(t) = -16t^2 + 148t equal to zero and solving for t.

Start with the equation:
-16t^2 + 148t = 0

Step 1: Factor out the common factor -4t:
-4t(4t - 37) = 0

Now, there are two possibilities:
1) -4t = 0 --> t = 0
2) 4t - 37 = 0 --> 4t = 37 --> t = 37/4

Step 2: Solve for t in each case:

1) When -4t = 0, t = 0. This means that at t = 0 (when the rocket is initially launched), the rocket is at ground level.

2) When 4t - 37 = 0, we can solve for t by isolating the variable:
4t = 37
t = 37/4

So, the rocket will take 37/4 seconds (or 9.25 seconds) to return to the ground.

To double-check our work, we can substitute t = 9.25 into the original equation to verify that h(t) equals zero:

h(9.25) = -16(9.25)^2 + 148(9.25)
= -16(85.5625) + 1369.5
= -1369 + 1369.5
≈ 0

As h(9.25) approximately equals zero, this confirms that the rocket will indeed return to the ground after approximately 9.25 seconds.

Please note that when working with equations, it's important to carefully follow the steps and double-check the work to minimize errors.

To find the time it takes for the rocket to return to the ground, we need to solve the equation h(t) = 0 because the height of the rocket will be zero when it reaches the ground. Substituting h(t) = 0 into the equation, we get:

0 = -16t^2 + 148t

Now, let's solve this quadratic equation step by step:

Step 1: Set the equation equal to zero:
-16t^2 + 148t = 0

Step 2: Factor out common terms if possible:
t(-16t + 148) = 0

Step 3: Apply the zero product property:
t = 0 or -16t + 148 = 0

Step 4: Solve for t in the second equation:
-16t + 148 = 0
-16t = -148
t = -148 / (-16)
t = 9.25

The values we found are t = 0 and t = 9.25. However, time cannot be negative in this context, so we discard t = 0.

Therefore, the rocket will take approximately 9.25 seconds to return to the ground.

Here are the math laws used in the solution:
- Zero product property: If a * b = 0, then a = 0 or b = 0.
- Factoring: Expressing a quadratic equation as a product of two binomials.
- Solving linear equations: Isolating the variable to find its value.

Please note that the answer has been rounded to two decimal places.