Question
A toy rocket is launched from a platform 2.1 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+14.0t + 2.1. How long will it take for the rocket to hit the ground?
Answers
GPT 3.5
To find the time it takes for the rocket to hit the ground, we need to find the value of t when the height, h, is equal to zero.
The equation is: h = -4.9t^2 + 14.0t + 2.1
Setting h to zero: 0 = -4.9t^2 + 14.0t + 2.1
This equation is a quadratic equation, and we can solve it by factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula:
The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions for x are: x = (-b ± √(b^2 - 4ac)) / (2a)
In our equation, a = -4.9, b = 14.0, and c = 2.1.
t = (-14.0 ± √(14.0^2 - 4(-4.9)(2.1))) / (2(-4.9))
Simplifying this equation, we get:
t = (-14.0 ± √(196.0 + 40.92)) / (-9.8)
t = (-14.0 ± √(236.92)) / (-9.8)
Now, we have to consider both the positive and negative solutions since time cannot be negative:
t = (-14.0 + √(236.92)) / (-9.8) or t = (-14.0 - √(236.92)) / (-9.8)
Simplifying these equations, we get:
t ≈ 4.13 seconds or t ≈ 0.312 seconds
Therefore, it will take approximately 4.13 seconds for the rocket to hit the ground.
The equation is: h = -4.9t^2 + 14.0t + 2.1
Setting h to zero: 0 = -4.9t^2 + 14.0t + 2.1
This equation is a quadratic equation, and we can solve it by factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula:
The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions for x are: x = (-b ± √(b^2 - 4ac)) / (2a)
In our equation, a = -4.9, b = 14.0, and c = 2.1.
t = (-14.0 ± √(14.0^2 - 4(-4.9)(2.1))) / (2(-4.9))
Simplifying this equation, we get:
t = (-14.0 ± √(196.0 + 40.92)) / (-9.8)
t = (-14.0 ± √(236.92)) / (-9.8)
Now, we have to consider both the positive and negative solutions since time cannot be negative:
t = (-14.0 + √(236.92)) / (-9.8) or t = (-14.0 - √(236.92)) / (-9.8)
Simplifying these equations, we get:
t ≈ 4.13 seconds or t ≈ 0.312 seconds
Therefore, it will take approximately 4.13 seconds for the rocket to hit the ground.