The height of an arrow shot upward can be given by the formula [tex]$s=v_0 t-16 t^2$[/tex], where [tex]$v_0$[/tex] is the initial velocity and t is time. How long does it take for the arrow to reach a height of 48 ft if it has an initial velocity of [tex]$96 ft / s$[/tex]? Round to the nearest hundredth.
The equation that represents the problem is [tex]$48=96 t-16 t^2$[/tex]. Solve [tex]$16 t^2-96 t+48=0$[/tex]
Complete the square to write [tex]$16 t^2-96 t+48=0$[/tex] as [ ] . Solve [tex]$(t-3)^2=6$[/tex]. The arrow is at a height of 48 ft after approximately [ ] s and after [ ] s.
Answer (2)
Solve the equation ( t − 3 ) 2 = 6 by taking the square root of both sides, which gives t − 3 = ± 6 .
Solve for t to get t = 3 ± 6 .
Calculate the two values of t : t 1 = 3 + 6 ≈ 5.449 and t 2 = 3 − 6 ≈ 0.551 .
Round the values to the nearest hundredth to find the times when the arrow is at 48 ft: 0.55 s and 5.45 s.
Explanation
Problem Analysis Let's analyze the problem. We are given the height of an arrow as a function of time, s = v 0 t − 16 t 2 , where v 0 is the initial velocity. We are given that the initial velocity v 0 = 96 f t / s , and we want to find the time t when the height s = 48 f t . This leads to the equation 48 = 96 t − 16 t 2 . We are asked to solve the equation 16 t 2 − 96 t + 48 = 0 by completing the square, and we are given that the equation can be written as ( t − 3 ) 2 = 6 .
Solving for t We are given the equation ( t − 3 ) 2 = 6 . To solve for t , we take the square root of both sides:
( t − 3 ) 2 = ± 6
t − 3 = ± 6
t = 3 ± 6
Calculating the values of t Now we calculate the two possible values for t :
t 1 = 3 + 6
t 2 = 3 − 6
Using a calculator, we find that:
t 1 ≈ 3 + 2.449 = 5.449
t 2 ≈ 3 − 2.449 = 0.551
Rounding to the nearest hundredth, we get:
t 1 ≈ 5.45
t 2 ≈ 0.55
Final Answer Therefore, the arrow is at a height of 48 ft after approximately 0.55 seconds and after approximately 5.45 seconds.
Examples
Imagine you're launching a model rocket straight up into the air. Knowing the initial speed of the rocket, you can use the quadratic equation to determine how long it will take to reach a certain height. This is useful for timing experiments or ensuring the rocket doesn't exceed a safe altitude. Similarly, in sports like archery or basketball, understanding projectile motion helps athletes optimize their shots for distance and accuracy. By calculating the trajectory, they can adjust their angle and velocity to hit the target effectively.
After solving the equation, the arrow reaches a height of 48 ft at approximately 0.55 seconds and again at approximately 5.45 seconds.
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