sketch-the-indicated-curves-by-the-methods-of-this-section-you-may-check-the-graphs-by-using-a-cal-9

Question

Sketch the indicated curves by the methods of this section. You may check the graphs by using a calculator.
The altitude $h$ (in $\mathrm{ft}$ ) of a certain rocket is given by $h=-t^{3}+54 t^{2}+480 t+20,$ where $t$ is the time (in s) of flight. Sketch the graph of $h=f(t)$

Jonathon Brumley · Accepted Answer

we have a baseball pitcher throwing a ball from a height of six feet toward a plate to 60 feet away, and he&#x27;s gonna release the ball with a speed of 130 feet per second. We want to determine the height of the ball as it crosses the flip as a crosses the plate. The vertical position formula is going to be negative 16 t square plus visa bough T. But the original vertical velocity is zero, so we don&#x27;t need to worry about that middle term and then plus y Sebo, which is the initial height, which would be six feet. The horizontal position function x of T off would eat well. 130 the horizontal velocity times the time it&#x27;s traveling now, in order to figure out the height of it as a crosses the plate, we&#x27;re gonna have to figure how long it takes this ball to travel 60 horizontal meters. Now, on your calculator. If you were to graph those Parametric equations, you get a graph that looks something like this. It&#x27;s half of a parabola. Start to the height of six, and when it&#x27;s 60 feet away, you&#x27;re gonna see that its above the ground. Okay, first for the time that it takes to travel 60 meters, we&#x27;re just going to go to the horizontal motion formula. I&#x27;m gonna fill in 64 x empty. Since that the distance that we wanted to travel, I will divide 60 by 130 which wouldn&#x27;t give me point 462 So it takes 0.462 seconds to reach the plate. So now let&#x27;s figure the height of it. At that point, I will fill in 0.462 seconds for the time. So I have negative 16 times 160.462 squared, plus six fourth vertical position. And when we feel that in, we get a high of 2.5 eight feet. So the pitches up 2.58 feet above the plate when it crosses

Linh Vu · Answer

would you use on, like, Africa critter conduct this mosque and what you need to intend a function now first with the brackets. Yeah. And then Xbk agility Power three and my last June. They you got You know why? We would come on now. And why did they want U minus three day? Um, from here, we need to send, uh, interval for that B s A diggers for minus 10. Detained now and then we have a photograph. Looks like this one on. And we try to extend diggers from minus infinity. Infinity. You have a full craft. Looks like this one.

Charles Carter · Answer

I guess the prepare A, um our domain. It would make sense for just started. Zero. How many do some work from? See what? It hits the ground. So this graph, she can look like a problem. Uh, you know, hit the ground at some point. So is in a set of equal to zero. Since then, our equation equal to zero zero equals negative. 16 t square plus 90 60 plus 80. We went about everything by negative. 16. What? A T square minus 16. Find a spot. This isn&#x27;t gonna be very pretty. So magically quadratic formula s a minus. Bi a six points or minus. The square root of 36 minus four times a temp C. It&#x27;s 20. Want to? Yeah, Okay. Says six, uh, plus or minus four when were going toe to pauses. Miss something. Okay, we got my discriminate wrong. That&#x27;s where it is. Okay, so it&#x27;s plus 20. Makes a big difference. So it&#x27;s not plus or minus four. I knew messed up because, like the route here, uh, would have been negative. And six plus or minus four does not give me negative. So we need six plus or minus the square root of 56 which is definitely our thing. So we subtracted. That&#x27;s gonna give us a negative side to get our positive value. We just have to add. So we&#x27;re gonna get ah, 56 coming. Pull four out of that. Okay. We sure camp. So our answer is gonna be, um our domain will go till three. Ah, Plus. And I said we could pull a four out. Certainly left with 14. So three, plus the root of 14. Thank you. And we could round that off to be kind of probably close to 6.8. No, no, no. Just estimating. But that&#x27;s the exact answer right there. Okay. Part being. Okay. So to reach the highest point, we&#x27;re just gonna use our Vertex formula. That&#x27;s gonna be a minus. Bi over to a So that time would be at home. We can use our same rounded part. Ah, well, let&#x27;s just do 96 will have negative 96 over negative 32 which is just gonna be three seconds till it reaches its maximum and the high at that time, we seem to plug in three into our equation. The negative 16 times three square plus 96 times three plus 80 Can that high is gonna be equal. Teoh 224. Thank you.

Jose Hannan · Answer

Hello, everybody. In this video, I&#x27;ll be showing you how to solve Exercise 31 in Chapter one, Section one of calculus early transit dentals that this problem tells us that we have a vertically flying rocket is high as a function of time is given by negative 16 t squared. Let&#x27;s 90 60 plus 80 with T representing time and seconds and starting AT T equals zero the time that the rocket first lifts off. The first part of this problem asked us to determine an appropriate domain for this function. That is what values of T makes sense for this problem. Clearly, we can&#x27;t go below t equals zero, as if that is when the rocket starts its flight. This interval also ends when the rocket hits the ground and stops flying. So the first thing we want to do is find the zeros of this function to determine when the rocket hits the ground we set each of T equal to zero means that we have zero equals negative 16 t squared. Let&#x27;s 90 60 let&#x27;s 80 To simplify things. Let&#x27;s divide this whole thing by negative 16. What this gives us is T squared minus 60 minus five zero. Unfortunately, this can&#x27;t really be factored that easily. So we&#x27;re gonna need the quadratic formula when we use the formula. What we get is that T equals negative B. Which year is negative? Six. We have positive six plus or minus. He squared, which is 36 minus four times a, which is one time see, which is negative. Five quantity is square. Read it. That is all divided by two way. Which year is just to Tim You. We know we have six plus or minus 36. Let&#x27;s 20 which is that product under the radical all over, too. And that stuff is just 56. No notice that 56 is equal to four times 14 so we can square it for and pull it out of the radical. What we get this six plus or minus to Route 14 over to. And of course, if we can&#x27;t let the factor of two, what we get is three Lester minus Route 14. However, as for the re three minus Route 14 notice that 14 is greater than three squared and less than foreswear, and therefore the square root of 14 is gonna be greater than three. So we can say that this difference is listen. Zero. Since the rocket starts AT T equals zero, we can neglect this answer as it does not fit into the context of our problem and therefore a rocket. It&#x27;s the ground at the Value three, plus Route 14 and therefore inappropriate domain for this function is given by zero when the rocket takes off and three plus Route 14 2nd which is when the rocket hits the ground. And that&#x27;s a side note. Notice that this is about six point 74 So that is how you solve part of the problem. Now. Part B of this question assets to graph the function h of tea and from this graph determined the time at which the rocket reaches its maximum height, as well as what this maximum height is at this time. So that is exactly what I&#x27;ve done here. On here is a graph of H of tea and because of the application on which I&#x27;m graphing this, we&#x27;re able to directly get information about the next month that it occurs at T equals three seconds, and the rocket achieves a high of 224. So I really doubt that this problem is asking you to painstakingly graph dysfunction by hand and from that graph fined the maximum and that they&#x27;re probably expecting you to do this on a computer. So this is most likely the exact method that they want you to dio. And for such case, that is all you need to do for part of the problem. However, for the purposes of this video, I wanted to show you ah, more complete alternative way of finding this maximum if you&#x27;re so interested, which takes advantage of the properties of a parabola, most notably its symmetry. If you recall from earlier we found the two routes for this parable which were a three plus route 14 and three minus route 14 because of parabolas symmetric. The point on the TXs which corresponds to its vertex, which is the point where it reaches its maximum height. That time is going to be the midpoint between these two routes. So let&#x27;s find this midpoint to do this. We have three minus Route 14 and we add three plus Route 14 the other route that we divide this by two we see that the root Fourteens cancel. We get six over to which is three. And just as we had found before on the graph, this is the time when the rocket reaches its maximum height. Now, to find the height, all we have to do is plug three into the original function, and what we get is the age of three equals negative. 16 times three squared, which is nine plus 96 times three. Let&#x27;s 80 16 times nine is 1 44 We have negative 1 44 plus 96 times three, which is to 88. Let&#x27;s 80. From here we get that this is 1 44 plus 80 and that is a sequel to 200 and 24 feet. And again, this is exactly what we found by plotting the graph and seeing what the computer told us. So again, it doesn&#x27;t seem like you have to do this for this problem, but it is just a more complete, more manual way of solving the problem without using a computer. So anyway, that is all for exercise. 31

Linh Vu · Answer

would you use on, like, Africa? Corretta Colder decimals and change into the function here. But the axe is good unity, don&#x27;t you? Minus one and funding why we have it. The my nose far be square on. Then we can send it interval apology saying this for months. Stan, you know that on. And it would be the crab. Looks like cat. So maybe you consent to Rome minus, uh, fit. Then let&#x27;s say, you know, fit the S well way. Except so it doesn&#x27;t have the left hand side. Only this time from minus one Father accident gets after infinity here so that we were a little far.

Problem

Sketch the indicated curves by the methods of thi…

Problem 51 Easy Difficulty

Sketch the indicated curves by the methods of this section. You may check the graphs by using a calculator.
The altitude $h$ (in $\mathrm{ft}$ ) of a certain rocket is given by $h=-t^{3}+54 t^{2}+480 t+20,$ where $t$ is the time (in s) of flight. Sketch the graph of $h=f(t)$

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Allyn J. Washington, Richard S. EvansBasic Technical Mathematics with Calculus

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus