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Ifa stone is thrown down at 120 ft/sec from by 8 (t) = 1000 height of 1000 feet; its height $ after 120t 16t? with feet seconds is given Compute $' (€) a...

Question

Ifa stone is thrown down at 120 ft/sec from by 8 (t) = 1000 height of 1000 feet; its height $ after 120t 16t? with feet seconds is given Compute $' (€) and hence findthe stonesvelocity at timest = 0. 1.2 3 and 4 seconds: (4)(0)8 (1)(2}(2)When docs the stonc rcach the ground how fast is it travcling when it hits the ground? (Hint: It reaches the ground wnen the helzht Is zero_ other words when -The stone hlts the ground alterHecanatThestone OravelingIusecthen hita dhe Erald

Ifa stone is thrown down at 120 ft/sec from by 8 (t) = 1000 height of 1000 feet; its height $ after 120t 16t? with feet seconds is given Compute $' (€) and hence findthe stonesvelocity at timest = 0. 1.2 3 and 4 seconds: (4) (0) 8 (1) (2} (2) When docs the stonc rcach the ground how fast is it travcling when it hits the ground? (Hint: It reaches the ground wnen the helzht Is zero_ other words when - The stone hlts the ground alter Hecanat Thestone Oraveling Iusecthen hita dhe Erald



Answers

A stone is dropped off a 256 -ft cliff. The height of the stone above the ground is given by the equation $h=-16 t^{2}+256$ where $h$ is the stone's height in feet, and $t$ is the time in seconds after the stone is dropped $(t \geq 0)$ Find the time required for the stone to hit the ground.

The given quadratic equation is X equals two minus 16 T square plus 2 56. Now we have to find the time required for the stone to hit the ground here. The condition is that that equals to zero. So we have to put that equals to zero which gives zero equals two minus 16 p square plus 2 56. Now factor out sGC F. So we have zero equals two minus 16. A square minus 16. Now factories difference of square. So we have zero equals two minus 16. T plus four, multiplied by a minus four. Now set each factor equals to zero. So we have minus 16 equals to zero. This is not our solution. Now put a plus four equals to zero which gives equals two minus four. We know that time cannot be negative. 30 minus four plus four is not our solution. Now put a minus four equals to zero. It gives two equals to four. So the time required for the restaurant to hit the ground is four centimeters and this is our final answer. Sorry second.

At one case an object. It's from thrown from the top of the way such that it has a parabolic look. Another kids, an object is throwing from the crowd. Look, some public. We want to discover the value of peace of Syria so that both items are at the highest point. So first thing we walked into, we want to find the same. That's my point. We first need to discover when does this happen? So to find out when it happens, we want to find the first or evidence of of teeth. With respect to t years in the summit difference, roll one with the power room two times negative 16 as negative 32. Rewrite the base of T. Subtract one from exponents of tune to leave us with the export of one plus 32 because the derivative of 32 t is 32 finally the derivative of a constant zeros of turkey for 48 to seal it Next we want to set this tribute of equal is your in tow her now, when the project was reached, its maximum in the first situation. So gonna solve when negative 32 t equals unable to dividing both sides. By negative 32 we obtain the T equals one second. Next, we wanted to the same thing for the second object. So we have a position function J t we're gonna find when this function will reach its maximum by finding the derivative of Jean with respect to team firms remembering that figs zero is a constant. So the driven of a native 16 teas clear multiplying the to the front, rewriting the base, subtracting one from the exponents we obtain negative 32 t again and the derivative of a constant times of variable will give us just the constant with these subzero again to find when the maximum Wilco, we're gonna set this motion equal, just hero inside for teeth. So I have a negative 32 2 equals negative being subsidies dividing both sides by negative 20 to negative divided by negative. There's a positive. So I have these subzero delighted with 32 seconds. So that's now when the Maxwell Makos, So to actually find maximum right three trunk shin, we're going to substitute thes times into their corresponding functions. So to find the maximum right for half of tea performing to evaluate the function f number one second. So again we're substituting into the original function. So have negative 16 times. Well, one quantity squared plus 32 terms. One plus 48. We're going to simplify this to obtain native 16 plus 32 plus 48 poor 64 feet. The second function will reach its maximum height. Always substitute the time which will reach a maximum height into the function. So again, we're gonna and put the time into the function. So we want G v sub zero divided by 32. So came in putting the T value entirely. Original function Back tune made of 16 times the quantity Big subsidiary over 32 Quantity square plus being subzero Times are time which is getting some serum divided by 32. Since we want the heights to be the same, we're going toe also input 64 feet for that word Jake function. So the next we saw when 64 equals neighbor 16 times the subzero virgin quantity squared plus B sub zero times these obscure over 32 which means simplifying the right side. So piece of zero squared is the subzero squared we have times the negative 16 and 32 square. I'm gonna catch the right as 32 squared for now. Obviously, we could multiply 32 times, three to which will do a little bit later and then multiplying three across in the second time, not saying these ABS deals were provided for two. What if next find the common? Did that denominator on the right side? The common denominator would be 32 square. So we multiply both sides of the equation by 32 squared, which at this point going in and most playoff. But that ISS, which would be 1000 and 24. So multiplying both sides of the equation by 1024 remembering that that means to distribute 1024 toe each turn on the right side. So 64 times 1024 on the left side will give us 65,000 536. 1020 Forward times. Nate. The first term on the right side. The project aim the family. The 32 square, which was 1024 in the 1024 will reduce to leave this negative 16 times that b squared. Sorry times be sincere Square and then 32 devised until 1024 32 times so from that term were left with 32 times he subsea of squared. Next, we're gonna combine flight terms on the right side. Negative 16 These subzero squared plus 32 the subzero square. We'll get us a positive 16 season. Feel square still equal in that 65,536 more left side now to solve for the Zip zero square, we're going to divide both sides by 16 from memory deuce. So we obtain that piece of Zeer squared. It's 4000 96. Let's remind ourselves that we had a moment that went up and down, so our initial velocity was positive. So, keeping in mind that we need visas here to positive we're now going toe, apply the square root property, which technically would say to take the pleasure minus square lead over 4096. But since we know that big sub zeros positive, we can just take the positive square root for 4096 he saw for being subsidiary. And that's where little 4096 will be 64. So, as a result, our initial velocity also notice that would be subsidiary would be 64 feet percent.

Okay, so we're asked to find our maximum height of giving our fallen functions so we'll start by taking or derivative. I'm finding are critical points. So we haven't got to t plus 64 will set this equal to zero and solve it you. So we get 32 t is equal to 64 so t is equal to two. Okay, so we want to find our maximum height. So we need to also include our endpoints. So we'll check to use people to zero to use it with the to and to use it to six. Now, let's put this into our original function s city and find our output value. So at T zero, we got 100 and 92. 20 is, um, equal to two. He gets 256 on when t is able to six yet So Okay, so we see that the maximum height is 256. And this is when our T value is to

Defense and the question is going that is strong in is thrown stepped out. Found the roof of a building support. It is a building. Okay. And a strong is true. Now I want to check out from the roof, the height of the stronger than an empty. Yeah. And any attempt to measure from the ground is given by. Was it from the ground? This is the ground bench. Hft attempted. Okay one second. It is the function is given that at ft. Is it close to one is 16 years square. The 60 40 Plus 80. So he will differentiate it. So it will be first to minus of 30 people the 64. So and ye Htut close to zero B B. B. Question who's speaking creditable stg question monastery that is negative. So we can see that. Had a question to it has a maximum hide so you can see that. Hey question too john Harris maximum. Hi So edge of to that in the 1st 2- of 16 did we spare less 60 400 to blessed be. So this man is 16- four. Yes Sorry it has been 20. The city one is 64 plus 1 28 plus 80. So it will be for us to Mhm. 1 44. Mhm. Mhm. So this is the maximum heart of the strong Kalinda. No. Okay so this is done so yeah. Yeah. Yeah. I hope your initial. Thank you.


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