a rocket launched off a cliff is represented by the equation h(t)= -16t^2+100t+140. when will this rocket reach 200 feet?
I tried to factor it, i'm confused.
-16t^2 + 100t + 140 = 200.
-16t^2 + 100t - 60 = 0.
Divide by 4:
-4t^2 + 25t - 15 = 0.
Use Quadratic Formula.
Solutions: T1 = 0.67 s, T2 = 5.8 s.
No
To find when the rocket reaches 200 feet, we need to solve the equation h(t) = 200.
Starting with the equation h(t) = -16t^2 + 100t + 140, we substitute h(t) with 200:
-16t^2 + 100t + 140 = 200
Rearranging the equation to set it equal to zero:
-16t^2 + 100t + 140 - 200 = 0
Combining like terms:
-16t^2 + 100t - 60 = 0
To solve this quadratic equation, we can either factor, complete the square, or use the quadratic formula. In this case, let's use factoring.
We need to find two numbers that multiply to give -16 * -60 = 960 and add up to 100.
The two numbers that satisfy these conditions are 120 and -8.
Rewriting the equation:
-16t^2 + 120t - 8t - 60 = 0
Factoring by grouping:
-8t(2t - 15) -2(2t - 15) = 0
Now we can factor out the common term:
(2t - 15)(-8t - 2) = 0
Setting each factor equal to zero and solving for t:
2t - 15 = 0 or -8t - 2 = 0
2t = 15 or -8t = 2
t = 15/2 or t = -2/8
Simplifying:
t = 7.5 or t = -1/4
Since time cannot be negative in this context, we discard the negative solution.
Therefore, the rocket will reach 200 feet at t = 7.5 seconds.
To find when the rocket reaches a height of 200 feet, we need to solve the equation h(t) = 200. In this case, the equation is h(t) = -16t^2 + 100t + 140.
To solve it, we need to set -16t^2 + 100t + 140 equal to 200:
-16t^2 + 100t + 140 = 200
To simplify the equation, let's move 200 to the left side of the equation:
-16t^2 + 100t + 140 - 200 = 0
Now, combine like terms:
-16t^2 + 100t - 60 = 0
Since the equation is in the form of a quadratic equation, we can solve it using the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
For the equation -16t^2 + 100t - 60 = 0, the coefficients are:
a = -16
b = 100
c = -60
Substituting these values into the quadratic formula:
t = (-100 ± √(100^2 - 4(-16)(-60))) / (2(-16))
Simplifying further:
t = (-100 ± √(10000 - 3840)) / (-32)
t = (-100 ± √(6160)) / (-32)
Since we are talking about time, we can discard the negative solution:
t = (-100 + √(6160)) / (-32)
Performing the calculations:
t ≈ -0.92 or t ≈ 8.42
Since time cannot be negative in this context, we disregard the negative solution. Thus, the rocket will reach a height of 200 feet approximately 8.42 seconds after launch.