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From a 128-foot tree, an object is thrown straight up into the air then follows a trajectory. The height S(t) of the ball above the building after t seconds is given by the function S(t) = 96t - 16t^(2). How long will it take the object to reach maximum height?

User Shuo
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1 Answer

15 votes
15 votes

Answer:

3 seconds

Explanation:

We can solve this in either of two ways: Graphing or taking the first derivative, I'll use both,

Graphing

Plot the equation. My DESMOS graph is attached. One can find the vertex of this curve at 3 seconds. Bonus: It reaches a height of 144 units at this point.

Derivative

The first derivative of this function will produce an equation that returns the slope of the line at any point x.

S(t) = 96t - 16t^(2)

S'(t) = 96 - 2*16t

S'(t) = 96 - 32t

The slope of the curve will be 0 when the ball reaches it's maximum height and begins to fall. Since we want the time it takes to reach a slope of zero, we can set S'(t) to 0 and solve:

S'(t) = 96 - 32t

0 = 96 - 32t

32t = 96

t = 3 seconds

From a 128-foot tree, an object is thrown straight up into the air then follows a-example-1
User Gbitaudeau
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