Item 20 You are riding a roller coaster when a shoe falls off your foot. The function y=200−16t2 represents the height y (in feet) of the shoe t seconds after it falls off your foot. The shoe lands on the top of a 31-foot-tall building. After how many seconds does the shoe hit the building?

Assuming you mean y = 200 - 16t^2, we have all the required information needed to solve this problem. The y-value is the height of the building and the t-value represents the number of seconds after the shoe fell off.

Since we are trying to solve for t, we will be using our y-value. The problem states that we landed on a building with a height of 31 feet. We can plug this into the y-value, since that is what y is defined as (the height of the building).

Now we have:
31 = 200 - 16t^2

We can solve this to find t:
-169 = -16t^2
169 = 16t^2
10.5625 = t^2
3.25, -3.25 = t

We have found two answers for t. However, the negative value is not a solution because we can not have a negative number of seconds. Thus, 3.25 is the only value of t which works for this problem.

Since t is the value we are trying to find, we have our answer. The shoe hits the building after 3.25 seconds.