A ball is thrown upward with an initial velocity of 14 meters per second from a cliff that is high. The height of the ball is given by the quadratic equation
H = 49t^2 + 14t +60
where h is in meters and t is the time in seconds since the ball was thrown. Find the time that the ball will be 30 meters from the ground. Round your answer to the nearest tenth of a second.
Sorry, the cliff is 30 feet high.
To find the time that the ball will be 30 meters from the ground, we need to solve the quadratic equation for H = 30.
The equation given is: H = 49t^2 + 14t + 60
Substituting H with 30: 30 = 49t^2 + 14t + 60
Rearranging the equation to bring it to the standard quadratic form (ax^2 + bx + c = 0): 49t^2 + 14t + 60 - 30 = 0
Simplifying: 49t^2 + 14t + 30 = 0
Now, we can use the quadratic formula to find the values of t:
t = (-b ± √(b^2 - 4ac)) / (2a)
In this equation, a = 49, b = 14, and c = 30.
Substituting the values: t = (-(14) ± √((14)^2 - 4(49)(30))) / (2(49))
Simplifying further: t = (-14 ± √(196 - 5880)) / 98
Calculating inside the square root: t = (-14 ± √(-5684)) / 98
Since the value inside the square root is negative, it means that there are no real solutions for t. This implies that the ball will not reach a height of 30 meters during its upward motion.
Therefore, the time at which the ball will be 30 meters from the ground is nonexistent.