The height h(t) in feet above the ground of a golf ball depends on the time, t (in seconds) it has been in the air. Ed hits a shot off the tee that has a height modeled by the velocity function h(t) = 0.5at^2 + vt + s where a is -32 ft/sec^2, v is the initial velocity, and s is the initial height.1. Write a function that models the situation.2. Sketch a graph of the function and describe the graph verbally.3. Identify the key components of the graph (minimum or maximum, vertex, roots) and give a brief description of what each represents. 4. What is the height of the ball at 1.5 seconds? Is there another time at which the ball is at the same height?5. At what time (s) will the ball be 48 feet above the ground?6. How long after the ball is in the air does it reach its maximum height?7. After how many seconds will the ball hit the ground?Please see attachment for the solution.MODELLING:The general modelling equation for the height at time is given by (1) (1)Where acceleration due to gravity, (in the vertical direction) and (tee height above the ground)The speed at which the ball is hit in a direction with the ground we assume is and thus by considering vectors we have a vertical component of velocityThus on substitution into the general model we get (2) (2)This model (2) describes the vertical height of the golf ball at any
subsequent time when driven at a velocity in any particular direction
with respect to the horizontal groundNow to try and describe a typical situation we need to assume some initial conditions. Let us consider that the golf ball is driven at an angle with respect to the horizontal ground and at an initial velocity ofThen the vertical component of velocity is Let us also consider that the golf ball initially sits on the tee a height Then our modelling equation becomes that given by (3)Since (3)See a graph I have drawn below [1] of vertical height against time based on this modelVertex as shown point co-ordinates when maximum heightRoot as shown when ball hits the ground againMaximum when maximum heightMinimum is obviously when ball at the groundCOMPUTATIONS:Height of the ball at time can be computed by substituting in equation (3) and working out , when we do thisHeight of golf ball after 1.5s is thus There is another time when the golf ball is at this height of 213.35ft and we find this time by solving the equation (4) below for its roots for Thus we need to solve (4)Using the general form for solving a quadraticWhere specifically in (4) we have and we get roots (the time we already know)Or when the ball is at the same heightTo find the times when the ball is 48ft above the ground we solve the quadratic for with Using (3)Again using the quadratic formula to solve for where and we solve forThus at timesAndWe can thus say that the golf ball will be 48ft off the ground at times of and To find the times for the maximum height we need to compute the function describing the maxuimum/minimum which is just when the differential of our model function (3) is zeroLooking at our model function (3) again (3)The differential is given byEquating this to zero we getAnd solving for we getThis is the time that the ball reaches maximum heightTo determine the time that the ball hits the ground we solve the model equation (3) for when thusUsing the quadratic formula to solve for and we solveThus the time that the balls hits the ground is the higher root and whenOr rounded when the ball hits the groundReferences:

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