Evaluating and Solving Quadratic Functions Duffer McGee stood on a hill and used

Question

Evaluating and Solving Quadratic Functions Duffer McGee stood on a hill and used a nine iron to hit a golf ball that reached a maximum height of 133 feet and stayed in the air for 5.4 seconds before it touched the ground. Pretty good for a Duffer. 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: 4 H(t)107169 219257283 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 Preview Based on your model how high is the hill from which the golf ball was hit?? The golf ball was hit from a hill Use your model to estimate how long the golf ball will take to reach its maximum height and what its feet high. maximum height will be. Round your answers to two decimal places. The golf ball will reach a maximum height of feet after seconds Use your model to determine how long it will take for the golf ball to hit the surface of Mercury Round vour answer to two decimal places. The golf ball will reach the surface of Mercury after seconds.

Explanation / Answer

To find the quadratic regression for the above given function of height w.r.t. to time -

Let's take

h(t)=  at^2+bt+c

Now putting given values of t in the above equation we will get-

h(1)= a+b+c=107. (A)

h(2)= 4a+2b+c=169. (B)

h(3)= 9a+3b+c=219 . (C)

Now we have 3 equation and 3 variables

From equation (A)

c= 107-(a+b)

Put it (B) and (C) we will get-

3a+b=62.

8a+2b=112.

By solving these 2 equation we will get

a= -6, b=80 and putting these in the value of c

c=33

The required quadratic regression model value is -

so h(t)= -6*t^(2)+80t+33.

Now to find the value of golf ball to reach it's maximum height in given time

One needs to differentiate the given function w.r.t 't' as it will give the slope of tangent to h(t) curve w.r.t. time (t).

dh(t)/dt=-6*2 t+80

Now at the highest point where the ball will hit the slope of the curve should be zero.

dh(t)/dt= -12t+80=0

t= 80/12=6.66seconds

And to find the height at this instant

Put this value into given value of h(t)

h(6.66)= -6*(6.66*6.66)+80*6.66+33

h(6.66)= 299.664 feet.

Now one needs to find total time of flight when the golf ball will hit the surface.

For that we need to put the solution of the equation h(t)=0.

-6t^(2)+80t+33=0

By solving this quadratic equation we will get valid value of

time t = 13.733 seconds.

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