When a golfer strikes a golf ball represent its vertical height, h, after t seconds using the following representatioN

Question

When a golfer strikes a golf ball and we observe the ball’s trajectory, we can represent its vertical height, h, measured in feet above the ground after t seconds using the following representation h(t) = 96t − 16t 2 . a. Find the height of the ball after 4 seconds. Is the golf ball moving upward or downward after 4 seconds? b. Evaluate the rate of change of h(t) on the time interval [3, 3.5]. Interpret the meaning of your answer and be sure to include appropriate units. c. Find the maximum height of the golf ball. d. State a realistic domain for the function h(t). e. Sketch the function h(t) and label when the ball is moving upward, moving downward, and the point where the golf ball is at its highest point (include the coordinates of the highest point)

Answer 1

we have h(t)=96t-16t²

it is a projectile motion.

I am using concept of derivative to find the maximum hight at the value of t

so first derivative of the given function is

96-32t

put 96-32t=0

t=3sec.

now second derivative it will be -32 which is negative so at t=3 function is maximum.

a) at t=4

height will be

96×4-16×4²= 384-256

=128m

b) first derivative is the rate of change it means

(h(3.5)-h(3))÷0.5

140÷0.5

280m

c) Maximum height will be at t=3

so h(t)=96×3-16×3²=288-144=144m

d) h(t)=0

96t-16t²=0

t=0,6

that implies that at t=6 ball will be at the ground

so the domain of the given function will be [0,6]

projectile motion