An object is dropped from the top of a 400-ft building. The height h of the ball after t seconds is modeled by the equation h=16f^2+400. How long is the ball in the air?
To find how long the ball is in the air, we need to find the time it takes for the height of the ball to reach 0.
Given the equation h = 16t^2 + 400, we set h = 0 and solve for t:
0 = 16t^2 + 400
Dividing both sides of the equation by 16:
0 = t^2 + 25
Rearranging the equation:
t^2 = -25
Since time cannot be negative, there are no real solutions for t in this case.
Therefore, the ball is not in the air for any amount of time.
To find out how long the ball is in the air, we need to determine the time it takes for the ball to hit the ground. The ball will hit the ground when its height, h, becomes zero.
Given the equation h = 16t^2 + 400, we can set it equal to zero and solve for t:
16t^2 + 400 = 0
Dividing both sides by 16:
t^2 + 25 = 0
Subtracting 25 from both sides:
t^2 = -25
At this point, we can see that the equation has no real solutions because we cannot take the square root of a negative number. Therefore, the ball will never hit the ground, which means it will stay in the air indefinitely.
To determine how long the ball is in the air, we need to find the value of t when the height h is equal to zero. At the highest point of the ball's trajectory, the height will be zero because it will be on its way back down.
Given the equation h = 16t^2 + 400, we can set h to zero and solve for t:
0 = 16t^2 + 400
Now, let's solve the equation:
Subtract 400 from both sides:
-400 = 16t^2
Divide both sides by 16:
-25 = t^2
Taking the square root of both sides:
t = ±sqrt(-25)
Since time cannot be negative in this physical scenario, we discard the negative value. Thus, the ball is in the air for t = sqrt(-25) seconds.
However, it is important to note that the square root of a negative number is not a real number. Therefore, in this particular case, the equation does not have a real solution. This means that the ball does not spend any time in the air based on the given equation.
It's possible that there may be additional factors or constraints that are not included in the given information, so it's always a good idea to double-check the problem statement or consult the accompanying context to ensure all relevant information is considered.