If a rocket is propelled upward from ground level, its height in meters after t seconds is given by h=-9.8t^2+88.2t. During what interval of time will the rocket be higher than137.2m?
Do I set it up like this?
-9.8(1)^2+88.2(1)+137.2 = 215.6
Then h=b/2a=9.8/215.6=22m
137.2+22=159
Thanks for your help.
Not quite. You found (sort of) when the height is greatest (the vertex of the parabola). That was not the question.
You have
h=-9.8t^2+88.2t
when is h=137.2?
(Aside. You sure about that 9.8? h = vt - 1/2 at^2 and a = 9.8)
137.2 = -9.8t^2 + 88.2t
t = 2,7
So, at those two times, h=137.2
Knowing what you do about the shape of parabolas, and that this one opens downward, h > 137.2 between those two values, so
2 < t < 7
To determine the interval of time during which the rocket will be higher than 137.2m, you need to solve the given equation for t when h is greater than 137.2m.
So let's set up the equation:
-9.8t^2 + 88.2t > 137.2
To solve this inequality, we need to rearrange it into the form: "ax^2 + bx + c > 0" and find the values of x that satisfy the inequality.
-9.8t^2 + 88.2t - 137.2 > 0
Now, we can solve this quadratic inequality by factoring, completing the square, or by using the quadratic formula. In this case, let's use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / 2a
Plugging in the values for this equation:
t = (-88.2 ± √(88.2^2 - 4(-9.8)(-137.2))) / (2(-9.8))
Now we can calculate the values of t using a calculator. We get two solutions:
t ≈ 1.68 seconds
t ≈ 9.05 seconds
These are the points in time when the rocket will be at a height of 137.2m. However, we need to find the interval of time during which the rocket will be higher than 137.2m.
To determine this, we need to consider the intervals between these two points. There are two cases to consider:
1. Before t ≈ 1.68 seconds: The rocket has not reached a height of 137.2m yet.
2. Between t ≈ 1.68 seconds and t ≈ 9.05 seconds: The rocket is higher than 137.2m.
3. After t ≈ 9.05 seconds: The rocket has descended below a height of 137.2m.
Therefore, the interval during which the rocket is higher than 137.2m is approximately 1.68 seconds < t < 9.05 seconds.