A toy rocket is launched form a 3m platform, at 8.2m/s. The height of the rocket is modelled by the equation h=-4.9t^2+8.1t+3 where h is the height, in metres, above the ground and t is the time, in seconds.

After how many seconds will the rocket rise to a height of 6m above the ground?

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To find out after how many seconds the rocket will rise to a height of 6m above the ground, we need to substitute the given height value into the equation and solve for t.

The equation that models the height of the rocket is:
h = -4.9t^2 + 8.1t + 3

We want to find the time (t) when the height (h) is 6m. So we substitute h = 6 into the equation:
6 = -4.9t^2 + 8.1t + 3

Now we need to solve this quadratic equation for t. To do this, we rearrange the equation to bring all terms to one side:
-4.9t^2 + 8.1t + 3 - 6 = 0
-4.9t^2 + 8.1t - 3 = 0

Now we can either use factoring, completing the square, or the quadratic formula to solve this equation. Let's use the quadratic formula, which states that for a quadratic equation of the form ax^2 + bx + c = 0, the solution for x is given by:
x = (-b ± √(b^2 - 4ac)) / (2a)

In our case, a = -4.9, b = 8.1, and c = -3. Plugging these values into the quadratic formula:
t = (-8.1 ± √(8.1^2 - 4(-4.9)(-3))) / (2(-4.9))
t = (-8.1 ± √(65.61 - 58.8)) / (-9.8)
t = (-8.1 ± √6.81) / (-9.8)

Now we calculate the two possible solutions for t:
t1 = (-8.1 + √6.81) / (-9.8)
t2 = (-8.1 - √6.81) / (-9.8)

Using a calculator, we find:
t1 ≈ 0.659 seconds
t2 ≈ 1.335 seconds

So the rocket will rise to a height of 6m above the ground after approximately 0.659 seconds or 1.335 seconds.