Respuesta :
The equation for the height of the rocket at time t given
[tex] h= -16t^2+192t [/tex]
We have to find the time t, when the rocket reaches 560 feet.
That means we have to find t when h = 560 ft. we will place 560 in the place of h to find t now.
[tex] h= -16t^2+192t [/tex]
[tex] 560 = -16t^2+192t [/tex]
In the right side, we can check -16 is the common factor. So we will take out -16 from the rigbht side.
[tex] 560 = -16(t^2 - 12t) [/tex]
To get rid of -16 from the right side and move it to left side, we will divide both sides by -16.
[tex] 560/-16 = -16(t^2-12t)/-16 [/tex]
[tex] -35 = t^2 -12t [/tex]
Now we will move -35 to the righ side by adding 35 to both sides.
[tex] -35+35 = t^2-12t+35 [/tex]
[tex] 0 = t^2 -12t+35 [/tex]
[tex] t^2-12t+35 = 0 [/tex]
We will factorize thee left side to find the values of t now. We need to find a pair of factors of 35 that by adding them we will get -12.
The pair of factors of 35 are -5 and -7 and by adding -5-7 we will get -12.
[tex] t^2-12t+35 =0 [/tex]
[tex] (t-5)(t-7) =0 [/tex]
So by using zero product property we will get
[tex] t-5 =0 [/tex]
[tex] t-5+5 = 0+5 [/tex]
[tex] t=5 [/tex]
Also [tex] t-7 =0 [/tex]
[tex] t-7+7 = 0+7 [/tex]
[tex] t=7 [/tex]
So we have got the rocket reaches at 560ft when t = 5 seconds and also when t = 7 seconds.
Now part b.
When the rocket completes its trajectory and hits the ground then the height or h = 0. So we will place h = 0 there in the equation.
[tex] h= -16t^2+192t [/tex]
[tex] 0= -16t^2 + 192 t [/tex]
[tex] 0 = -16(t^2-12t) [/tex]
[tex] -16(t^2-12t) = 0 [/tex]
We will move -16 to the other side by dividing it to both sides.
[tex] -16(t^2-12t)/-16 = 0/-16 [/tex]
[tex] t^2-12t = 0 [/tex]
We will take out the common factor t from the left side. By taking out t we will get,
[tex] t(t-12) = 0 [/tex]
We will use zero product property now. By using that we will get,
[tex] t = 0 [/tex]
ans also [tex] t-12 = 0 [/tex]
[tex] t-12+12 = 0+12 [/tex]
[tex] t = 12 [/tex]
When the rocket completes its trajectory and hits the ground the time t can not be 0. When t =0, the rocket starts the trajectory.
So when the rocket completes its trajectory and hits the ground ,
then t = 12seconds.
So we have got the required answers.
