A model rocket is launched with an initial velocity of 212ft/sec. It's height h, in feet, after t seconds is given by the formula h=212t-16t^2. After how many seconds will the rocket first reach a height of 592ft?

To get the time, we just substitute h = 592 to the equation:

h = 212t - 16t^2
592 = 212t - 16t^2
16t^2 - 212t + 592 = 0
4t^2 - 53t + 148 = 0
Factoring,
(t - 4)(4t - 37) = 0
t = 4 s
t = 37/4 s
Since it requires the time in which the rocket FIRST reaches a height of 592, we choose the smaller, which is t = 37/4 s.

Hope this helps :3

*lol I'm sorry the smaller time is t = 4s. OMG XD

To find out after how many seconds the rocket will first reach a height of 592ft, we need to set the equation for height equal to 592ft and solve for t.

The given equation for height is h = 212t - 16t^2.

To find the value of t when h = 592ft, we replace h with 592 and solve the resulting equation:

592 = 212t - 16t^2.

We rearrange the equation to form a quadratic equation:

16t^2 - 212t + 592 = 0.

Now, we can solve this quadratic equation using factoring, completing the square, or the quadratic formula. The quadratic formula is generally the most straightforward method for solving any quadratic equation.

The quadratic formula is stated as:

t = (-b ± √(b^2 - 4ac)) / (2a),

where t is the variable we are solving for, and in our equation, a = 16, b = -212, and c = 592.

Applying the values to the quadratic formula, we get:

t = (-(-212) ± √((-212)^2 - 4 * 16 * 592)) / (2 * 16),

which simplifies to:

t = (212 ± √(44944 - 37888)) / 32.

Now, calculating the square root and simplifying further:

t = (212 ± √(706)) / 32.

t = (212 ± 26.57) / 32.

So we have two possible solutions for t:

1) t = (212 + 26.57) / 32,
2) t = (212 - 26.57) / 32.

Calculating these values, we find:

1) t ≈ 8.25 seconds,
2) t ≈ 6.53 seconds.

Therefore, the rocket will first reach a height of 592ft after approximately 8.25 seconds or 6.53 seconds.