Suppose you start at the point (0,0,3) and move five units along the curve x=3sint, y=4t, z=3cost in the position direction. Where are you now?
=(3sin1,4,3cos1)
Arclength is given by R0k |r 0(t)|dt, and in this case |r 0(t)| = 5, so we want to
find the k such that this integral is equal to 5, i.e., R0k 5dt = 5k, so k = 1. Thus we have
traveled 5 units along the curve when t = k = 1, so we’re at (3 sin 1, 4, 3 cos 1).
Arclength is given by R0k |r 0(t)|dt, and in this case |r 0(t)| = 5, so we want to
find the k such that this integral is equal to 5, i.e., R0k 5dt = 5k, so k = 1. Thus we have
traveled 5 units along the curve when t = k = 1, so we’re at (3 sin 1, 4, 3 cos 1).(R0k=integration from 0 to k)
Well, if I start at the point (0,0,3) and move five units along the curve x=3sint, y=4t, z=3cost in the position direction, I'm pretty sure I'll end up in a different spot. If I had to guess, I'd say I'll end up somewhere with different coordinates. But hey, who am I to judge? Maybe I'll end up at the exact same spot just with a different perspective. It's all relative, right?
To find the new position, we need to substitute the parameter value into the given parametric equations and travel along the curve in the position direction.
Given parametric equations:
x = 3sin(t)
y = 4t
z = 3cos(t)
Start point: (0, 0, 3)
We are moving 5 units along the curve in the position direction. This means we need to find the parameter value (t) such that the distance traveled is 5 units.
To do this, we can calculate the arc length of the curve and find the appropriate time parameter value.
1. Calculate the arc length of the curve:
The arc length formula for parametric curves is given by:
s(t) = ∫ √((dx/dt)^2 + (dy/dt)^2 + (dz/dt)^2) dt
Using the given parametric equations:
dx/dt = 3cos(t)
dy/dt = 4
dz/dt = -3sin(t)
Substituting these values into the arc length formula:
s(t) = ∫ √((3cos(t))^2 + 4^2 + (-3sin(t))^2) dt
Integrating this expression would give us the total arc length of the curve. However, in this case, we can use a different approach.
2. Simplify the arc length expression:
Notice that the coefficients of dx/dt, dy/dt, and dz/dt are all constant values. We can factor them out to simplify the expression.
s(t) = ∫ √(9(cos(t))^2 + 16 + 9(sin(t))^2) dt
= ∫ √(9(cos(t))^2 + 9(sin(t))^2 + 16) dt
= ∫ √(9((cos(t))^2 + (sin(t))^2) + 16) dt
= ∫ √(9 + 16) dt
= ∫ √25 dt
= ∫ 5 dt
Integrating 5 with respect to t gives:
s(t) = 5t + C
where C is the integration constant.
3. Calculate the parameter value for the distance traveled:
We know that the distance traveled is 5 units. So we can set s(t) = 5 and solve for t.
5t + C = 5
t = (5 - C)/5
Here, t represents the parameter value at which the distance traveled is 5 units. The integration constant C depends on the initial position and the chosen path.
4. Find the corresponding (x, y, z) coordinates:
Substitute the calculated parameter value into the given parametric equations.
x = 3sin(t) = 3sin((5 - C)/5)
y = 4t = 4((5 - C)/5)
z = 3cos(t) = 3cos((5 - C)/5)
Plug this value of t into the equations above to find the new position coordinates (x, y, z).
Put t=5
then
(0.26, 20, 2.98)