Hit the volleyball

[misspigtails]

Carl Allan hit the volleyball at 3 ft above the ground with an initial velocity of 32 ft/sec. The path of the ball is given by the function S(t) = -16t² + 32t + 3, where S is the height of the ball at t seconds. What is the maximum height reached by the ball?

 

[HallsofIvy]

Carl Allan hit the volleyball at 3 ft above the ground with an initial velocity of 32 ft/sec. The path of the ball is given by the function S(t) = -16t² + 32t + 3, where S is the height of the ball at t seconds. What is the maximum height reached by the ball?

Complete the square: S(t)= -16t²+ 32t+ 3= -16(t+ a)^2+ b. What are a and b. Do you see that, because -16 times a square is never positive, S(t) is always less than or equal to b?

 

[erblina]

Carl Allan hit the volleyball at 3 ft above the ground with an initial velocity of 32 ft/sec. The path of the ball is given by the function S(t) = -16t² + 32t + 3, where S is the height of the ball at t seconds. What is the maximum height reached by the ball?

Firstly take S(t)=-16t² +32t+3 as a function and then find the maximum value of S(t) using derivatives .
First derivative :
S’(t)=-32t+32, now look where S’(t)>0 and S’(t)<0
-32t+32>0 implies t<1
-32t+32<0 implies t>1
That means for t=1 the first derivative changes the sign and it’s the maximum value of S(t) , to be sure we prove the second derivative in t=1 if its negative means that the point t=1 is maximum value of S(t)
Second derivative :
S’’(t)=-32t at the point t=1 we have S’’(1)=-32 or S’’(1)<0
In the end the ball reaches maximum in time t=1s with height S=19.

 

[lookagain]

Firstly take S(t)=-16t² +32t+3 as a function and then find the maximum value of S(t) using derivatives .

erblina, that would be done in the calculus section.

One of the typical intermediate/advanced algebra methods is to find the number of seconds where

the maximum occurs as t = -b/(2a), where S(t) = at^2 + bt + c.

a = -16
b = 32

Solve for t.

Then substitute that t-value everywhere there is a t into -16t^2 + 32t + 3 to get the number of feet
where the ball reaches its maximum height.