Trinity is throwing a tennis ball up in the air from the ground. Trinity is 5 feet tall, and is throwing the ball straight up with an initial speed of 64 feet per second.

The function: f(t) = −16t 2 + 64t + 5, where t represents time in seconds, models this situation.

Trinity’s friend Max is in an apartment building that is next to where Trinity is throwing the ball. He is taking action pictures of the ball from his window. Max is able to take pictures when the height of the ball is more than 50 feet above Trinity.

Use the information here and the given function above to write an inequality to represent this problem situation. (Hint: If Trinity is 5 feet tall, what would 50 feet above her be measured from the ground?)

Determine the time period (t) range when Max will be able to take pictures of the ball. (Hint: graph your inequality as two different equations, find their intersection and interpret the results.)

Max wants to take a picture when the ball reaches its maximum height.
What is the Maximum height of the ball? Explain how you found the max height.
Maximum height:

How did you find the maximum height:

At What time (t) should Max take a picture of the ball? (HInt: What is the t value at your maximum height)

Consider that the window in the building is 55 feet above the ground. What is the vertical distance between the Max in the window and the maximum height of the ball?

Trinity decides to throw another ball in the air right after the first ball. She wants to throw it when the first ball is less than 61 feet in the air, but not on the ground zero feet.
Use the information here and the original given function above to write an inequality to represent this problem situation. (Hint: You need and upper bound and lower bound to your inequality)

Determine the time period (t) ranges when Trinity should throw the second ball. (Hint: graph your inequality as two different equations, find their intersection and interpret the results.)

You should have two different time periods that Trinity could throw the second ball. Is any time period more reasonable than another? Explain your reasoning.