Select the correct answer.
Terrance kicks a soccer ball at his game. Solve the given equation to find the times, [tex]$t$[/tex], when the ball lands back on the ground, in seconds.
[tex]$-16 t^2+40 t=0$[/tex]
How many seconds is Terrance's ball in the air?
Answer (2)
The ball is in the air for a total of 2.5 seconds. This was determined by solving the quadratic equation − 16 t 2 + 40 t = 0 , which gave us the two time solutions of t = 0 and t = 2.5 . Thus, the relevant time when the ball lands back on the ground is at t = 2.5 .
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Factor the quadratic equation: − 16 t 2 + 40 t = − 8 t ( 2 t − 5 ) = 0 .
Solve for t by setting each factor to zero: − 8 t = 0 or 2 t − 5 = 0 .
Find the two solutions: t = 0 and t = 2.5 .
Determine the time the ball is in the air: 2.5
Explanation
Understanding the Problem We are given the equation − 16 t 2 + 40 t = 0 , which models the height of a soccer ball as a function of time, t . We want to find the times when the ball lands back on the ground. This occurs when the height is 0. So we need to solve the equation for t .
Factoring the Equation To solve the equation − 16 t 2 + 40 t = 0 , we can factor out a common factor of − 8 t from both terms: − 8 t ( 2 t − 5 ) = 0
Solving for t Now, we set each factor equal to zero and solve for t :
− 8 t = 0 or 2 t − 5 = 0
For the first equation, − 8 t = 0 , we divide both sides by − 8 to get:
t = 0
For the second equation, 2 t − 5 = 0 , we add 5 to both sides to get:
2 t = 5
Then, we divide both sides by 2 to get:
t = 2 5 = 2.5
Interpreting the Solutions The two solutions for t are t = 0 and t = 2.5 . The solution t = 0 represents the time when Terrance initially kicks the ball. The solution t = 2.5 represents the time when the ball lands back on the ground.
Finding the Time in the Air The time the ball is in the air is the difference between these two times: 2.5 − 0 = 2.5 seconds. Therefore, Terrance's ball is in the air for 2.5 seconds.
Examples
Understanding quadratic equations like the one in this problem can help in various real-world scenarios, such as calculating the trajectory of projectiles, designing parabolic mirrors, or optimizing the dimensions of structures. For example, if you're designing a water fountain, you can use a quadratic equation to model the path of the water stream and ensure it lands where you want it to. Similarly, engineers use these equations to calculate the path of a ball thrown by a robot, ensuring it reaches its target accurately. The equation − 16 t 2 + 40 t = 0 is a simplified model, but it captures the basic principle of how gravity affects the motion of an object.
