A model rocket is launched off the top of a table. The height of the rocket is given by the function, h(t) = - 4.9t2 + 9.5t + 1.2 where h(t) is the height in metres and t is the time in seconds. How long will it take for the rocket to reach the ground?
When it reaches the ground, h(t) = 0
-4.9t^2 + 9.5t + 1.2 = 0
don't think it will factor, so let's use the formula
I got t = 2.0578 or a negative, the latter does not make sense
It will hit the ground after appr 2.06 seconds
To find the time it takes for the rocket to reach the ground, we need to find the value of t when the height, h(t), is equal to 0.
Given:
h(t) = -4.9t^2 + 9.5t + 1.2
Set h(t) to 0:
0 = -4.9t^2 + 9.5t + 1.2
This is a quadratic equation. We can solve it by factoring, completing the square, or by using the quadratic formula.
Let's use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
In this equation, a = -4.9, b = 9.5, and c = 1.2.
Substitute these values into the quadratic formula:
t = (-9.5 ± √(9.5^2 - 4(-4.9)(1.2))) / (2(-4.9))
Calculating the values inside the square root:
t = (-9.5 ± √(90.25 + 23.52)) / (-9.8)
Simplifying inside the square root:
t = (-9.5 ± √113.77) / (-9.8)
Taking the square root:
t = (-9.5 ± 10.66) / (-9.8)
Simplifying:
t = (-9.5 + 10.66) / (-9.8) or t = (-9.5 - 10.66) / (-9.8)
Calculating the two solutions:
t = 1.16 / (-9.8) or t = -20.16 / (-9.8)
Simplifying further:
t ≈ -0.118 or t ≈ 2.056
Since time cannot be negative in this context, we can discard the negative value of t.
Therefore, it will take approximately 2.056 seconds for the rocket to reach the ground.
To find the time it takes for the rocket to reach the ground, we need to find the value of t when the height, h(t), is equal to zero.
The equation represents a quadratic function, h(t) = - 4.9t2 + 9.5t + 1.2. To find when h(t) = 0, we set the equation equal to zero:
- 4.9t2 + 9.5t + 1.2 = 0
To solve this quadratic equation, we can use the quadratic formula:
t = (- b ± √(b^2 - 4ac))/(2a)
In our equation, a = -4.9, b = 9.5, and c = 1.2. Substituting these values into the formula, we get:
t = (-9.5 ± √(9.5^2 - 4(-4.9)(1.2)))/(2(-4.9))
Simplifying further:
t = (-9.5 ± √(90.25 + 23.52))/(2(-4.9))
t = (-9.5 ± √113.77)/(-9.8)
Now, we have two possible solutions for t:
t1 = (-9.5 + √113.77)/(-9.8)
t2 = (-9.5 - √113.77)/(-9.8)
Calculating these values using a calculator, we find:
t1 ≈ 1.124 seconds
t2 ≈ 1.018 seconds
Both t1 and t2 are positive values, which means they represent valid times. However, since the rocket is launched from the top of the table, it will take approximately 1.018 seconds for the rocket to reach the ground.