The equation h=-16t^2+112t gives the height of an arrow, shot upward from the ground with an initial velocity of 112ft/s,where t is the time after the arrow leaves the ground. Find the time it takes for the arrow to reach a height of 120ft.
Put in h for 120, rearrange it into quadratic form, and solve. I recommend the quadradratic equation.
To find the time it takes for the arrow to reach a height of 120 ft, we can set the equation h = 120 and solve for t.
The equation h = -16t^2 + 112t represents the height of the arrow as a function of time. We can substitute h with 120 and solve for t:
120 = -16t^2 + 112t
Now, let's rearrange this equation into the quadratic form:
-16t^2 + 112t - 120 = 0
To solve this quadratic equation, we can use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a),
where a, b, and c are the coefficients of the quadratic equation.
In this case, a = -16, b = 112, and c = -120.
Plugging these values into the quadratic formula:
t = (-112 ± √(112^2 - 4 * -16 * -120)) / (2 * -16)
Simplifying further:
t = (-112 ± √(12544 - 7680)) / -32
t = (-112 ± √(4864)) / -32
t = (-112 ± 69.72) / -32
Now, using the positive and negative solutions:
t1 = (-112 + 69.72) / -32 ≈ -1.11
t2 = (-112 - 69.72) / -32 ≈ 5.59
Since time cannot be negative, we discard the negative value of t.
Therefore, the time it takes for the arrow to reach a height of 120 ft is approximately 5.59 seconds.