The function f(t)=−8(t^2)+25 models the height of a ball dropped from 25 feet in the air.What was the speed of the ball when it was traveling the fastest (instantaneous rate of change)? Round all answers to three decimal places.The best method of solving will receive Brainliest!
Accepted Solution
A:
The ball will be fastest right before it hits the ground because of gravity.
First, let's find the value of t where the ball hits the ground.
-8t²+25 = 0
-8t² = -25
t² = [tex] \frac{25}{8} [/tex]
t = [tex]\frac{5}{2\sqrt{2}} [/tex]
Note: t has to be positive, which is why there is no positive/negative symbol.
Now, let's calculate the derivative of this function. The derivative will give us the speed of the ball for time t.
d/dx = -16t
Now, let's plug in the value of t
-16 ( [tex]\frac{5}{2\sqrt{2}} [/tex] }
= [tex]\frac{-5(8)}{\sqrt{2}} [/tex]
= [tex] -20 \sqrt{2} [/tex]
Take the absolute value of that because speed can't be negative
[tex]20\sqrt{2} [/tex]
That's the speed of the ball. You can convert that into decimal if you'd like.