help with algebra problem

Question

Duffer McGee stood on a hill and used a nine iron to hit a golf ball that reached a maximum height of 149 feet and stayed in the air for 5.8 seconds before it touched the ground. 

Mercury has a gravity of approximately 12 feet per second squared compared to Earth's 32 feet per second squared. NASA did a simulation to try to determine how high the golf ball would fly and how long it would stay in the air on Mercury if it was hit at the same height, angle and velocity as Duffer's. The data below represent the results of that simulation: 

t	1	2	3	4	5
H(t)	112	181	238	283	316

Use the Quadratic Regression feature of your calculator to generate a mathematical model for this situation. Write the function below. Round each coefficient to the nearest whole number. 

h(t)=

Answer 1

I used Desmos Graphing Calculator to do a quadratic regression (ax² + bx + c) and got a nice fit. Looks like maximum height on Mercury would be 346.38 feet after 7.25 seconds. In the regression a = -6, b = 87 and c = 31.

Can you duplicate with another method?

Answer 2

on earth

h(t) =-16t^2 +vot

v(t) = h'(t)=-32t+vo = 0

t=5.8/2= 2.9 seconds = time to reach max height =vo/32

v0=32x2.9= 92.9 ft/s= initial vertical speed

t=5.8 time in the air

h(2.9)=-16(2.9)^2 +92.9(2.9)=134.85= max height

on mercury

h(t)=-(12/2)t^2+vot

v(t) =-12t +vo=0

t = vo/12 = time to max height= 92.9/12=7.741666...seconds

vo= 12t = initial vertical speed=92.9 ft/sec

h(vo/12) = -6(vo/12)^2 +vo(vo/12)

=-vo^2/24+vo^2/12

= vo^2/24 =92.9^2/24 = about 359.6 feet = max height

t=vo/6= time in the air= 92.9/6= 15.48333... seconds

h(1)=-6+92.9= 86.9 ft in 1 sec

simulated h(1)= 112

h(2)=-24+185.8=161.8 ft in 2 sec

simulated h(2)=181

h(3)=-54+278.7= -224.7 ft in 3 sec

simu;ared h(3)=238

h(4)=-96+375.6= 279.6 ft in 4 sec

simu;ated h(4)= 283

h(5)= -150+464.5= 314.5 ft in 5 sec

compare simulated height =316