Let denote the rocket's position, velocity, and acceleration vectors at time .
We're given its initial position
and velocity
Immediately after launch, the rocket is subject to gravity, so its acceleration is
where .
a. We can obtain the velocity and position vectors by respectively integrating the acceleration and velocity functions. By the fundamental theorem of calculus,
(the integral of 0 is a constant, but it ultimately doesn't matter in this case)
and
b. The rocket stays in the air for as long as it takes until , where is the -component of the position vector.
The range of the rocket is the distance between the rocket's final position and the origin (0, 0, 0):
c. The rocket reaches its maximum height when its vertical velocity (the -component) is 0, at which point we have
The y-value of the vertex is positive 3, as shown by the +3 on the right hand side of the equation, and the x-value is -1, from the (x+1)^2 (remember, when the number is inside the brackets, flip the sign) The vertex would be (-1, 3)
If you are looking for a rigorous answer (calculus), we must find the mininum point of the equation: f(x) = (x+1)^2 + 3 f
f'(x) = 2(x+1) = 2x + 2
2x + 2 = 0
x = -1
f(1) = (-1 + 1)^2 + 3
f(1) = 0 + 3 = 3
(-1, 3)
Answer: See below
Explanation:
Formula: V = pi x r^2 x h
Pi = 22/7
r = 15 ft
h = 35 ft
=> V = 22/7 x 35 x 15^2
=> V = 22 x 5 x 15^2
=> V = 110 x 225
=> V = 24,750 ft^2
Answer:
X=9 and X=5
Step-by-step explanation:
So you want to get one side to equal zero so you can factor and find x. I recommend subtracting 14x and adding 45 to both sides so you don't have to deal with a negative quadratic.
X^2 - 14X +45 = 0
Now you can factor. Your looking for two factors that equal 45 and add to -14
All factors of 45:
1 and 45, 3 and 15, 5 and 9, -1 and -45,
-3 and -15, -5 and -9
So out of those combinations -5 and -9 both multiply to 45 and add up to -14 so these are our common factors
(X-5)(X-9)=0
X-5=0 add 5 to both sides X=5
X-9=0 add 9 to both sides X=9
Horizontal means X axiz. so 6 units on the x axis. The points are always placed (x,y) which is A. (1,6) and (7,6)