If a rocket is propelled upward from ground level, it's 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 than 137.2m?
h=9.8t^2+ 88.2t
I divided both sides by -9.8 which gave me
t^2=-9
But I can't remember what step is next please help. Thanks in advance:-)
The equation:
137.2 = -9.8t^2 + 88.2t
Divide both sides by -9.8 (whatever operation you do to one side of an equation you must do to the other side as well):
(137.2)/-9.8 = (-9.8t^2 + 88.2t)/-9.8
-14 = t^2 - 9
Set the equation equal to 0:
0 = t^2 - 9 + 14
Try to factor:
0 = (t - 7)(t - 2)
Set the factors equal to 0:
t - 7 = 0
t - 2 = 0
Therefore, the two solutions are t = 2 and t = 7.
Interval: 2 < t < 7
What is the maximum height of the rocket and how long does it take to reach that maximum?
To find the interval of time during which the rocket will be higher than 137.2m, you need to solve the quadratic equation \(h = -9.8t^2 + 88.2t\) when \(h > 137.2\).
First, subtract 137.2 from both sides of the equation:
\(h - 137.2 = -9.8t^2 + 88.2t\)
Now you have a quadratic equation:
\(-9.8t^2 + 88.2t - (h-137.2) = 0\)
You can rearrange the equation to get it in standard form:
\(-9.8t^2 + 88.2t - h + 137.2 = 0\)
The next step is to solve this quadratic equation for \(t\). You can use the quadratic formula:
\(t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
For this equation, the coefficients are:
\(a = -9.8\)
\(b = 88.2\)
\(c = -h + 137.2\)
Substitute these values into the quadratic formula and simplify to find the values of \(t\) when \(h > 137.2\).
To find the interval of time during which the rocket will be higher than 137.2m, we need to set up an inequality using the given equation.
First, let's rewrite the equation as:
h = -9.8t^2 + 88.2t
We want to find the values of t for which the rocket is higher than 137.2m, so we set up the inequality:
h > 137.2
Substituting the equation for h, we have:
-9.8t^2 + 88.2t > 137.2
To solve this inequality, we can first subtract 137.2 from both sides to isolate the quadratic expression:
-9.8t^2 + 88.2t - 137.2 > 0
Next, we can divide the entire inequality by -9.8 to simplify further:
t^2 - 9t + 14 > 0
To solve this quadratic inequality, we can factor it:
(t - 2)(t - 7) > 0
Now, we can consider the signs of the factors to determine the intervals where the inequality is true. This can be done by making a sign chart or considering each factor separately.
For the factor (t - 2), it is positive when t > 2 and negative when t < 2.
For the factor (t - 7), it is positive when t > 7 and negative when t < 7.
To satisfy the original inequality, we need both factors to be greater than zero.
This occurs in two intervals:
1) t > 7
2) 2 < t < 7
Therefore, the rocket will be higher than 137.2m during the time interval t > 7 and 2 < t < 7.