(Modeling)Solve each problem. See Example 3.Height of a Projectil...

Question

(Modeling)Solve each problem. See Example 3.Height of a ProjectileA projectile is launched from ground level with an initial velocity of v_0 feet per second. Neglecting air resistance, its height in feet t seconds after launch is given by s=-16t^2+v_0t. In each exercise, find the time(s) that the projectile will (a) reach a height of 80 ft and (b) return to the ground for the given value of v_0. Round answers to the nearest hun-dredth if necessary. v_0=96

Accepted Answer

Hello, today we're gonna be solving the following problem. So we are told that the height of a toy rocket in T seconds after its release from the ground is defined by Y is equal to negative 4.9 T squared plus V initial T. If the initial velocity is V initial is equal to 23 m per second, we want to determine the time that it takes a toy to reach a height of 14 m. Also, we want to determine how long it'll take to hit the ground. And we're going to approximate our answers to the nearest 100th. So let's go ahead and just draw a quick diagram of what is going on. So we have this rocket that we're launching upward and we want to determine the time it takes in order for this rocket to reach 14 m. And we also want to determine the time it takes for the rocket to hit the ground at 0 m. So notice that when the rocket reaches 14 m, it's going to reach it on two spots of the diagram. So if we solve the equation for 14 m, we should expect two answers. So let's go ahead and start with that. No. In order to solve for when the rocket reaches 14 m, we know that the rocket is going to be at a height of 14. So we're going to let y equal to 14. Furthermore, we know that the initial velocity of the rocket is 23 m per second. So we're going to let the initial equal to 23 m per second. Now, what we're going to do is take the values of Y and V initial and plug it in to the position function given to us. And that is going to give us 14 is equal to negative 4.9 T squared plus 23 T. And now, since we want to solve for T, we want to set this equation equal to zero. In order to do that, we're going to add 4.9 T squared to the left-hand side of the equation and subtract 23 T to the left hand side. And that is going to give us 4.9 T squared minus 23 T plus 14 is equal to zero. So what we have is a trinomial and we can go ahead and solve for T using the quadratic equation. The quadratic equation states that we have X is equal to negative B plus or minus the square root of B squared minus four times A times C all over two times A and A B and C are going to be the coefficients of our given trinomial. So A is going to be the coefficient of the first term which is going to be 4.9. In this case, B is the coefficient of the second term which is negative 23 and C is going to be the constant which in this case is 14. So plugging these values in to the quadratic equation is going to give us T is equal to negative B which is going to be negative negative 23 which is positive 23 plus or minus the square root of B squared, which is negative 23 squared minus four times A which is 4.9 times C which is 14 all over two times A which is two times 4.9. And now we're gonna go ahead and simplify 23 squared or negative 23 squared is going to give us 529. And for negative four times 4.9 times 14 is going to give us negative 274.4. So what we have is 23 plus or minus the square root of 529 plus 274.4 all over two times 4.9 which is going to give us 9.8. Now, the square root of 529 plus 274.4 is going to simplify to give us an approximate value of 16. So what we are left with is 23 plus or minus 16 divided by 9.8. And since this is a plus or minus statement, we can go ahead and split this up into two equations, we have one equation where T is equal to 23 plus 16 divided by 9.8. And we have another equation where T is equal to 23 minus 16 divided by 9.8. So let's go ahead and simplify both of the values on the left hand side. 23 plus 16 is going to give us 39 and 39 divided by 9.8 is going to give us approximate value of 3.98 and 23 minus 16 is going to give us seven and seven divided by 9.8 is going to give us an approximate value of 0.72. So what this means is that the total time that is going to take for the rocket to reach 14 m in height is going to be 3.98 seconds and 0.72 seconds. Now let's go ahead and figure out how long it'll take for the rocket to hit the ground at 0 m. So in this case here, we're going to let Y equal to zero, but we're going to keep the same initial velocity which is 23 and plugging this into the equation given to us is going to give us zero is equal to negative 4.9 T squared plus 23 T. Here, we can go ahead and factor out A T from the right hand side and that is going to give us zero is equal to T times the quantity negative 4.9 T plus 23. And using the zero property, we can set both of these factors equal to zero. So what we have is T is equal to zero and negative 4.9 T plus 23 is equal to zero. Now, in this case here, T cannot be zero because when time is equal to zero, that is when the rocket is getting launched. But we want to figure out how long it'll take for when the rocket hits the ground at the end of the travel. So T cannot be zero. So we're gonna go ahead and solve the right hand side in order to sol for T on the right hand side, we're going to subtract 23 to both sides of the equation. And that is going to leave us with negative 4.9 T is equal to negative 23. Now, finally, we're going to divide both sides of this equation by negative 4.9. And that is going to give us T is equal to approximately 4.69. And that is means that it's going to take approximately 4.69 seconds in order for the rocket to hit the ground. So just to summarize what we came up with, we solved that it's going to take approximately 0.72 seconds and 3.98 seconds for the rocket to reach 14 m in height. And it's going to take 4.69 seconds for the rocket to hit the ground. With that being said, the answer to this problem is going to be a, So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

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