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This problem will prepare you for the Concept Connection on page 854 . The figure shows one hole of a miniature golf course.a. Is it possible to hit the ball in a straight line from the tee $T$ to the hole $H ?$ b. Find the coordinates of $H$, the reflection of $H$ across $\overline{B C}$ c. The point at which a player should aim in order to make a hole in one is the intersection of $\overline{T H}$ and $\overline{B C}$. What are the coordinates of this point? (GRAPH CAN'T COPY)

We're doing with a golf ball here And we got our equation H. F. X. Equals negative 32 X scored over 130 squared flashbacks and we've got a bunch of parts to that. Um And we should say that X. Is the horizontal distance santa cat stance? Um H over here. This is the height. All right. So now per ace has determined the height of the Gulf laughter. It's traveled 100 ft. So if we plug in 100 for the horizontal titian's now, how much is that? I need to grab a calculator in order to do that figurine. When I type that in, Ain't you? over 100 is approximately 81 ft. So it's kind of about 81 ft. Um Or when 81 ft high if it's traveled 100 ft. So now what about 300 ft? Age 300. How much is that? We'll plug it into this river fishing this year. 300 and that was 300 and I got approximately 129.6 feet. Yeah. Okay. Now see Sadd's What is the age of 500? Well I replaced my 300 worth of 500 and or my 100's worth 500. And I got 26.63 Uh feet. Actually it's just called that 26.6 ft. Now, does they interpret this value? So I'll say after. Thank golf ball. Thanks. Gone 500 ft. For example, It is 26.6 ft in the air. No, okay part and D says well how far was a hit? I'm gonna actually like that part E in order to answer that question because I think that would be easier. So um it says use a graphing utility to graphics. So favorite graphing utility right here. Dez mose. So let's go ahead and tank that inch. And they go 32 X. Squared the vibe hoops Divide by 130 Squared. That's acts thank zooming out. Zooming out. So I gets it because fights fights fights oh let's say shit It's a maximum height of 132. We'll probably need to know that. But when did they hit the ground up? Let's take a head figure around right there. Yeah. So if I'm gonna answer d I'm just going to say that it 528 ft. And so we'll go ahead and graph that, wow let's go ahead and put some key points on there. So this was 528 00. This point right here the vertex 264 Com 132. And then it starts out at 00. So he grafted it. Yeah. F says using graphing until you determine the distance that the ball shroud when the height is 90 ft. So when Why that is 90. Well looks like we got to play since. So this is f uh wow 115 ft And 413 ft. Yeah the balance 90 ft in the air. Mhm. Can I be here and there are approximately heart and G. Says how far does the ball travel before reaching its maximum height? What is the maximum height? So in Saint he is. Yeah It travels 164 ft to reach. Uh huh max hi. Of 132 ft. Yes. Mhm. Okay. And each so it says that just the value in our tv. Well see when the ball travels need its maximum height. Well this is the maximum hate well the distance that goes and then its height at that distance. And so I think that that's that's so um that's the problem.

For this question were given a set of data points, and they are the path of a ball thrown where X is the horizontal distance. And why is the height we're asked to find a quadratic equation in the form Y equals a X squared plus BX plus C that contains these points. To do that, we can create a system of equations and sulfur are unknowns, which are a BNC. Yeah, And then I'll use matrices to solve the system of equations. So to create our system of equations, I'm going to use what we know. So the first thing we know is that when x zero y is five. So if I plug in zero for X into our quadratic equation, we get zero plus zero plus C equals five. So our equation is C equals five. The second data point is when X is 15. Why is 9.6 so plugging in 15 for X, we get to 25 a plus 15 b first see is 9.6 and our third data point is when X is 30 we get 900 a plus. 30 b plus C is 12.4. Now we have our system of equations, and we can put it into an augmented matrix to solve for a BNC. All right, I'm going to start with this equation. I want this one to be last, so I'm going to do 225 a plus 15 b plus one C is 9.6. My second equation will be 900. A plus, 30 b plus one C is 12.4, and my third equation will be zero a plus zero B plus one C is five now. I can use elementary row operations. Start eliminating some terms. I want this to be zero. So to do that, I'm going to multiply the first row by negative four. That will give me 225 in the first cell. So when I add it to Row two, okay, zero. Yeah, so my first row stays the same. My second row becomes to 25 times negative four, which is negative. 900 plus 900 0 negative. Four times 15 plus 30 is negative. 30 and negative four times one plus one is negative. Three negative four times 9.6 plus 12.4 is negative. 26. My third row will stay the same now to get this into row echelon form. I want this to be zero. I'm sorry, I want this to be one and I want this to be one. So I'm going to divide Row one by 2 25 and I'm going to divide row to buy negative 30 that will give US one 15. Divided by 2 25 is 250.66 Repeating one divided by 2 25 is 250.44 REPEATING 9.6 divided by 2 25 is 0.4266 Repeating My second row will have zero negative 30 divided by negative 30 is one negative three divided by negative 30 0.1 and negative 26. Divided by negative. 30 is 300.0 0.866 Repeating. My third row will stay the same. 0015 Now that I have my matrix and row echelon form, I can use back substitution to solve for A B and C. So my first column contains a coefficients. My second is the B, and my third is the sea. So this third row here already tells us that C equals five. My second row tells us that B plus 0.1 c equals 0.866 repeating. And if I plug in that C is five we can solve for B. So we get that. B is approximately 0.37. It's 0.366 repeating, and this first equation tells us that a plus 0.66 b plus 0.44 c equals 0.4266 And if I plug in what I know, which is that B is 0.37 C is five. I can solve for a and I get a equals negative 0.4 So those are the coefficients to our quadratic formula, and we get y equals negative 0.4 x squared plus 0.37 x plus five. So we can graph that, and it looks like a parabola. Now. The third part of our problem says to graphically approximate the maximum height of the ball and the point at which the ball hits the ground. The maximum height. The ball is approximately here. Remember that are why access is the height and the point at which the ball hits the ground. Is that the horizontal distance when the height is zero? So it's approximately here. We could approximate those at about 14 ft, maybe in 105 ft. To solve this analytically. The maximum height is going to be at the Vertex, which is when X equals negative B over two. A. We can plug in what we know, which is V as 0.37 and A is negative 0.4 and we get that. The maximum height is when the horizontal distance is 46.25 ft. So that's about here 46.25 But we want to know the maximum height, and so that's why. So what's why, when X is 46.25 we just plug 46.25 into X here, and we get that Y is approximately 13.56 beach. Now we can find the point at which the ball hits the ground. That's going to be the X intercept, and we can find that by setting the equation equal to zero, and finally we can use the quadratic equation to solve this for X. So X is going to be negative B plus or minus the square root a B squared, minus four ac over to a. And when we solve that for X, we'll get that X is approximately one. Oh, four point. I believe it is 0.47 So the horizontal distance would be one Oh, four point 5 ft when the ball hits the ground and those do match up with what we thought they were graphically.

Now we're onto the fun game of golf. I love to golf myself. I sure wish I could hit the ball the way this model shows. So and you have to remember that most often we talk about hitting the ball in yards and this is dealing with feet. So when the ball is X feet away from the player, we're going to determine with this model what the height of the ball is. And so if we plug in 100 or find h of 100 and when we plug 100 in here, we end up finding that the ball is 81 point, about 1 ft above the ground. If we plug in 300 when the ball is 300 ft away, the ball is now 129.6 ft above the ground. So again, that's 100 yards away, minus nearing hitting with the ground then and even with my driver and when the ball is 500 ft away, mine is on the ground. But when the ball is 500 ft away for this particular golfer, we can see that the ball is getting closer to the ground so about 6 ft 20 about 27 ft above the ground went 500 ft away. And then you want to look at a graph, and just from this data, I didn't do it. Make a table of values to get my window. But I went in. I didn't let the calculator make my window. I put in a window and I said, You know, I'll bet you the ball goes higher than this So I had the X coordinate go from zero to, and I figured by 600 ft it would have hit the ground. So I just set my window to go from 0 to 600 ft, and I scaled by one hundreds on my graph for and under the X scale and for why I happen to go upto 150. And I scaled by like fifties. And when I did, I got this nice graph and I did something like this and it came down. We know it's parabolic and it hits the ground. And so then, after we get this graph, we had to use the graph for to help us find I believe in part D. It asks you to determine when the height is 90. You want to figure out using the graph for so when the height is equal to 90 ft above the ground. You wanted to have the graph for help. You determine when that's gonna happen. So I went on and I plugged. I had my model in his wife's up. One is the equation. And then why siptu? I typed in 90 and I believe it was 90. We'll just do a quick check. Let me go over. I don't think it was 100. Um, but Barbara, Barbara, um, and 90 ft is correct. And so I had a graph at 90 and had it draw the line across at 90 and then you could trace along. But typically, when I find this, I use my Intersect feature with my calculator. That's much more accurate. The Intersect feature. And so go to second and calculate and the Intersect. And it will tell you first function second function and guess and just move. Your is closer to the two locations, and I confined this spot of intersection in this spot, and I got this one to be at about 115 0.1, about 115 ft away from the golfer. That's when the ball will be 90 ft above the ground and then it will get upto high, and then it will come back down. And this will be the other time when it's gonna be 90 ft above the ground. And that is when the ball is about 413 ft 4130.1 ft away from the golfer. And my ball doesn't usually even go that far so well. It will go that far, but not much further than that. So then we're as to use the calculator, and you can set up a table of values and put your function in and start your table of values to go along. I think they tell you by increments of 25 you want to find this. You can also use your second calculate feature about Max and you'll have toe left round and right bound, and then you don't even have to move it and get to guess. But we'll find that this maximum is going to take place when the golfer is 200 and 64 ft are the ball has traveled 264 ft, and then it will end up being at its maximum height of 132 ft so we could do some of this information graphically as well. But this was basically using your calculator to help us determine these values. So on to another problem.

Suppose the path of a golf ball hit off a tee can be given by this vector. Valued function. R T equals 80 75 minus 0.1 a see negative five t squared plus 80 t The coefficient A is used to describe how much draw or fade Booker slice. Uh, the golfer puts on the ball when he or she strikes it, and we're going to investigate this for three different values of A. And what we're looking for is how far along the Y axis because that's the major amount of motion. The small travels in each of those three cases. Well, the first case is when there is no variation at all. We're going to use an Alfa on a value excuse me of zero. So the first thing we need to do is figure out when the ball reaches the ground, and that's determined by looking at the X component, the Z component that tells us how far up and then back down the ball travels vertically, and we want to know when it reaches, Ah, height back to zero. So the first thing we're gonna do this figure that out So we want to know when negative five T Square plus 80 t equals zero. Factoring out a negative by from both sides, we have negative by T from both terms. Negative five T times T minus 16 equals zero. So you get solutions of T equals zero or T equals 16. After 16 seconds, the ball comes back and hits the ground. It's equal. Zero were first. When that starts, let's figure out what it's why Component of distance is 40 equal 16. And again, if we have a equals zero in the distance travelled, it's gonna be 75 minus zero time. 16 Plugged into T, and that's going to give us 1200 sheet, which is 40 yards, which is a very serious Dr very impressive show for Part B. We're supposed to investigate this with a slight slice a equals zero point to Well, the slice doesn't have any effect on its vertical distance. There's no factor of a to show. Appear so again the amount of time it's in the air. The Time t four to reach the ground is gonna be the same t equal 16 that we found in part A, although now we want to see that horizontal distance travel. We're not looking at the total distance based on how far away from the Y axis it's traveled. We're just looking for the distance, a long the Y axis. So here we're going to have 75 minus 0.1 times 0.2 all of that times 16. Well, this is going to be 75 minus 0.2 time 16 which is 74 points. 98 climb 16. And I've done this off screen and this gives us about 101,199 0.6 some feet, so we'll call that 0.7 feet. So a slight slice or fade doesn't have a huge change. Less than half a foot in the driving distance report. See, imagining a bigger slice the value of a of 2.5. Well, again, the time to reach the ground isn't impacted by that quantity. So we're still looking at the time at 16 so they hear the distance. Travel is gonna be 75 minus 0.1 times 2.5 time 16. That's going to give us 75 minus 0.25 climb 16 which is going to give us 74.75 time 16 and again, I've done that calculation off screen, and that gives us a value of 1100 96 feet that comes up about four feet shorter than without any slice at all.


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