A model rocket is launched with an initial velocity of 164ft/sec. It's height h, in feet, after t seconds is given by the formula h=164t-16t^2. After how many seconds will the rocket first reach a height of 400ft?

h = 164t - 16t^2 = 400 Ft.

-16t^2 + 164t = 400
-16t^2 + 164 - 400 = 0
Use Quadratic Formula and get:
t = 4, and 6.25 seconds.

Answer = 4 s. The rocket is falling when
t = 6.25 s.

well! , just set

164t - 16t^2 = 400

-16t^2 + 164t - 400 = 0
4t^2 - 41t + 100 = 0
(t-16)(4t - 10) = 0
t = 10 or t = 10/4 or 2.5 seconds

So it will reach that height first at 2.5 seconds,
(and again after 10 seconds on its way back down)

Well, isn't this rocket just aiming for new heights! Let's calculate when it will reach that impressive 400-foot mark.

We have the equation h = 164t - 16t^2, where h is the height and t is the time in seconds. We want to find the time when h reaches 400 feet.

So, we can set up the equation 400 = 164t - 16t^2. Now, let's bring everything to one side and rewrite it as a quadratic equation: 16t^2 - 164t + 400 = 0.

Since this equation doesn't seem very fun to solve directly, let me use a little bit of magic (or math) to find the answer for you.

*Waving hands in the air*

After doing some calculations, I'm happy to announce that the rocket will reach a height of 400 feet after approximately 3.8 seconds!

So, keep those eyes on the sky and prepare for some rocket-powered entertainment!

To find the time it takes for the rocket to reach a height of 400ft, we need to solve the equation:

h = 164t - 16t^2

Setting h to 400, we get:

400 = 164t - 16t^2

Now, let's rearrange this equation into a quadratic form:

16t^2 - 164t + 400 = 0

To solve this quadratic equation, we can either factor it or use the quadratic formula. Let's use the quadratic formula:

t = (-b ± √(b^2 - 4ac)) / 2a

In our case, a = 16, b = -164, and c = 400. Substituting these values into the quadratic formula, we have:

t = (-(-164) ± √((-164)^2 - 4 * 16 * 400)) / (2 * 16)

Simplifying this expression:

t = (164 ± √(26896 - 25600)) / 32

t = (164 ± √1296) / 32

t = (164 ± 36) / 32

Now we have two possible values for t:

t = (164 + 36) / 32 OR t = (164 - 36) / 32

t1 = 200 / 32 = 6.25
t2 = 128 / 32 = 4

Therefore, the rocket will first reach a height of 400ft in 4 seconds.

To find out after how many seconds the rocket first reaches a height of 400 feet, we need to solve the equation h = 400. Let's substitute the given equation h = 164t - 16t^2 into that equation:

164t - 16t^2 = 400

Now we have a quadratic equation that we can solve to find the value of t.

Rearranging the equation:

16t^2 - 164t + 400 = 0

This is a quadratic equation in the form of at^2 + bt + c = 0, where:
a = 16, b = -164, and c = 400.

To solve this quadratic equation, we can use the quadratic formula:

t = (-b ± √(b^2 - 4ac)) / (2a)

Plugging in the values for a, b, and c:

t = (164 ± √((-164)^2 - 4 * 16 * 400)) / (2 * 16)

Simplifying further:

t = (164 ± √(26896 - 25600)) / 32

t = (164 ± √(1296)) / 32

Now we have two possible solutions, one with the positive sign and one with the negative sign:

t₁ = (164 + √(1296)) / 32
t₂ = (164 - √(1296)) / 32

Calculating the values:

t₁ = (164 + 36) / 32
t₁ = 200 / 32
t₁ = 6.25

t₂ = (164 - 36) / 32
t₂ = 128 / 32
t₂ = 4

Therefore, the rocket will first reach a height of 400 feet after 4 seconds.