A rocket is launch from the top of an 40 foot cliff with an initial velocity of 140 feet per second, the height h of the rocket after t seconds is given by the equation h=16t^2+140t+40. How long after the rocket is launch will be 30 feet from the ground?

First of all the equation should say:

h = -16t^2 + 140t + 40

you want h = 30
30 = -16t^2 + 140t + 40
16t^2 - 140t - 10 = 0

t = (140 ± √20240)/32
= 8.8 or a negative, which we will reject

it would take about 8.8 seconds to reach a height of 30 ft (on its way down)

Why do rockets never listen to classical music?

Because they prefer to rock and roll!

Now, let's figure out how long it takes for this rocket to be 30 feet from the ground. We can do this by setting the equation for height (h) equal to 30 and solving for time (t). So, we have:

30 = 16t^2 + 140t + 40

Now, we just need to solve this equation for t. Shall I do that for you?

To find out how long it will take for the rocket to be 30 feet from the ground, we need to solve the equation h = 30.

Given: h = 16t^2 + 140t + 40.

Step 1: Set h equal to 30 and rearrange the equation:
16t^2 + 140t + 40 = 30.

Step 2: Subtract 30 from both sides of the equation:
16t^2 + 140t + 40 - 30 = 0.

Simplifying the equation:
16t^2 + 140t + 10 = 0.

Step 3: Divide the equation by 2 to simplify:
8t^2 + 70t + 5 = 0.

Step 4: We can solve this quadratic equation using the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / 2a.

For our equation, a = 8, b = 70, and c = 5.

t = (-(70) ± √((70)^2 - 4(8)(5))) / 2(8).

Simplifying the equation further:
t = (-70 ± √(4900 - 160)) / 16.

t = (-70 ± √(4740)) / 16.

Step 5: We need to find the positive value of t since we are interested in the time after the rocket is launched. Calculating the square root:
t = (-70 ± √(120 * 39)) / 16.

t = (-70 ± √(120 * 3 * 13)) / 16.

t = (-70 ± √((2^3 * 3 * 13 * 2))) / 16.

t = (-70 ± 2√390) / 16.

Step 6: Simplify the expression further:
t = (-35 ± √390) / 8.

Since we are only interested in the positive value of t, the rocket will be 30 feet from the ground approximately 1.19 seconds after being launched.

To find the time at which the rocket will be 30 feet from the ground, we need to solve the equation h = 30, where h is the height of the rocket as a function of time.

Given the equation for the height of the rocket as a function of time:
h = 16t^2 + 140t + 40

We can substitute h = 30 into the equation and solve for t:
30 = 16t^2 + 140t + 40

Now let's solve this quadratic equation:
16t^2 + 140t + 40 - 30 = 0
16t^2 + 140t + 10 = 0

Since this is a quadratic equation, we can use the quadratic formula:
t = (-b ± sqrt(b^2 - 4ac))/2a

In this equation, a = 16, b = 140, and c = 10.

Now let's substitute these values into the quadratic formula and calculate the value of t:
t = (-140 ± sqrt(140^2 - 4 * 16 * 10))/2 * 16
t = (-140 ± sqrt(19600 - 640))/32
t = (-140 ± sqrt(18960))/32

Calculating the square root of 18960, we get:
t = (-140 ± 137.75)/32

Now, let's calculate t by adding and subtracting the square root:
t1 = (-140 + 137.75)/32 = -2.25/32 ≈ -0.0703
t2 = (-140 - 137.75)/32 = -277.75/32 ≈ -8.6813

Since time cannot be negative in this context, we discard t1 = -0.0703.
Therefore, the rocket will be 30 feet from the ground approximately 8.6813 seconds after it is launched.