A toy rocket is launched from a platform 1.4 meters above the ground in such a way this it’s height , h (in meters) after t seconds is given by the equation h=-4.9t^2+9.1t + 1.4. How long will it take for the rocket to hit the ground?
To find out when the rocket will hit the ground, we need to determine the time at which the height, h, is equal to 0.
The equation given is h = -4.9t^2 + 9.1t + 1.4.
Setting h to 0, the equation becomes 0 = -4.9t^2 + 9.1t + 1.4.
To solve this equation, we can either factor it or use the quadratic formula. Let's use the quadratic formula:
t = (-b ± √(b^2 - 4ac))/(2a)
In this equation, a = -4.9, b = 9.1, and c = 1.4.
Substituting these values into the quadratic formula:
t = (-(9.1) ± √((9.1)^2 - 4(-4.9)(1.4))) / (2(-4.9))
Simplifying this equation:
t = (-9.1 ± √(83.41 + 27.44)) / (-9.8)
t = (-9.1 ± √110.85) / (-9.8)
Now we have two possible solutions:
t ≈ (-9.1 + √110.85) / (-9.8)
t ≈ (-9.1 - √110.85) / (-9.8)
Calculating these two solutions:
t ≈ (-9.1 + 10.53) / (-9.8)
t ≈ 1.43 / (-9.8)
t ≈ -0.146
t ≈ (-9.1 - 10.53) / (-9.8)
t ≈ -19.63 / (-9.8)
t ≈ 2
Since time cannot be negative in this context, the rocket will hit the ground after approximately 2 seconds.
To find the time it takes for the rocket to hit the ground, we need to determine when the height of the rocket, h, is equal to 0.
Given the equation for the height of the rocket as a function of time, h=-4.9t^2+9.1t + 1.4, we set h=0 and solve for t.
-4.9t^2+9.1t + 1.4 = 0
To solve this quadratic equation, we can use the quadratic formula: t = (-b ± √(b^2 - 4ac)) / (2a)
In this equation, a = -4.9, b = 9.1, and c = 1.4.
Plugging in the values into the quadratic formula, we get:
t = (-9.1 ± √(9.1^2 - 4(-4.9)(1.4))) / (2(-4.9))
Simplifying further:
t = (-9.1 ± √(82.81 + 27.44)) / (-9.8)
t = (-9.1 ± √(110.25)) / (-9.8)
t = (-9.1 ± 10.5) / (-9.8)
Now we can solve for both possibilities separately:
For t = (-9.1 + 10.5) / (-9.8):
t = 1.4 / (-9.8)
t ≈ -0.143
And for t = (-9.1 - 10.5) / (-9.8):
t = -19.6 / (-9.8)
t ≈ 2
Since time cannot be negative in this context, we can discard the negative value.
Therefore, the rocket will hit the ground approximately 2 seconds after it was launched from the platform.
To find out how long it will take for the rocket to hit the ground, we need to determine the value of t when the height, h, is equal to 0. In other words, we need to solve the equation -4.9t^2 + 9.1t + 1.4 = 0.
To solve this quadratic equation, we can use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
Where:
a = -4.9
b = 9.1
c = 1.4
Plugging in these values into the formula, we get:
t = (-9.1 ± √(9.1^2 - 4(-4.9)(1.4))) / (2(-4.9))
Simplifying further, we have:
t = (-9.1 ± √(82.81 + 27.44)) / (-9.8)
t = (-9.1 ± √110.25) / (-9.8)
Now, we have two possible answers for t:
t1 = (-9.1 + √110.25) / (-9.8)
t2 = (-9.1 - √110.25) / (-9.8)
Evaluating these expressions, we get:
t1 ≈ 2.196 seconds
t2 ≈ -0.015 seconds
Since time cannot be negative in this context, we discard the negative value. Therefore, the rocket will hit the ground approximately 2.196 seconds after it is launched.