The equation h = -16t^2 + 112t gives the height of an arrow, shot upward from the ground with an initial velocity of 112 ft/s, where t is the time after the arrow leaves the ground. Find the time it takes for the arrow toreach a height of 180 ft.

put in for h 180 ft, factor (you may have to use the quadratic equation), and solve for t.

We will be happy to critique your thinking, or work.

To find the time it takes for the arrow to reach a height of 180 ft, we need to substitute 180 ft for h in the equation h = -16t^2 + 112t and solve for t. Here's how we can do it:

1. Replace h with 180 ft in the equation:
180 = -16t^2 + 112t

2. Rearrange the equation to match a quadratic equation form by bringing all terms to one side:
-16t^2 + 112t - 180 = 0

3. To solve for t, we can either factor the quadratic equation or use the quadratic formula. In this case, factoring might be a bit complicated, so let's use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)

For our equation -16t^2 + 112t - 180 = 0, we have:
a = -16, b = 112, c = -180

4. Plug these values into the quadratic formula:
t = (-112 ± √(112^2 - 4(-16)(-180))) / (2(-16))

Simplifying further:
t = (-112 ± √(12544 - 11520)) / (-32)
t = (-112 ± √(1024)) / (-32)

5. Continue simplifying:
t = (-112 ± 32) / (-32)

Now, we have two possibilities:
t = (-112 + 32) / (-32) or t = (-112 - 32) / (-32)

6. Solve for each t value:
t1 = (-112 + 32) / (-32) = -80 / (-32) = 2.5
t2 = (-112 - 32) / (-32) = -144 / (-32) = 4.5

The time it takes for the arrow to reach a height of 180 ft is either 2.5 seconds or 4.5 seconds, depending on the direction you consider.