PO
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Answer:
a) There's a zero between [1,2]
b) There's a zero between [1.5,2]
c) There's a zero between [1.5,1.75].
Step-by-step explanation:
We have
A)We need to show that f(x) has a zero in the interval [1, 2]. We have to see if the function f is continuous with f(1) and f(2).
We can see that f(1) and f(2) have opposite signs. And f(1)>f(2) and the function is continuous, this means that exists a real number c between the interval [1,2] where f(c)=0.
B)We have to repeat the same steps of A)
For the subinterval [1,1.5]:
f(1) and f(1.5) have the same signs, this means there's no zero in the subinterval [1,1.5].
For the subinterval [1.5,2]:
f(1.5) and f(2) have opposite signs, this means there's a zero between the subinterval [1.5,2].
C)We have to repeat the same steps of A)
For the subinterval [1,1.25]:
f(1) and f(1.25) have the same signs, this means there's no zero in the subinterval [1,1.25].
For the subinterval [1.25,1.5]:
f(1.25) and f(1.5) have the same signs, this means there's no zero in the subinterval [1.25,1.5].
For the subinterval [1.5,1.75]:
f(1.5) and f(1.75) have opposite signs, this means there's a zero between the subinterval [1.5,1.75].
For the subinterval [1.75,2]:
f(1.75) and f(2) have the same signs, this means there isn't a zero between the subinterval [1.75,2].
The graph of the function shows that the answers are correct.
The equation for the height of the rocket at time t given
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.
In the right side, we can check -16 is the common factor. So we will take out -16 from the rigbht side.
To get rid of -16 from the right side and move it to left side, we will divide both sides by -16.
Now we will move -35 to the righ side by adding 35 to both sides.
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.
So by using zero product property we will get
Also
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.
We will move -16 to the other side by dividing it to both sides.
We will take out the common factor t from the left side. By taking out t we will get,
We will use zero product property now. By using that we will get,
ans also
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.