Answer:
Angle = Ф =
(0) = 0
Hence, it is proved that angle between position vector r and acceleration vector a = 0 and is it never changes.
Step-by-step explanation:
Given vector r(t) = ![e^{t}cost i + e^{t}sint j](https://tex.z-dn.net/?f=e%5E%7Bt%7Dcost%20i%20%2B%20e%5E%7Bt%7Dsint%20j)
As we know that,
velocity vector = v = ![\frac{dr}{dt}](https://tex.z-dn.net/?f=%5Cfrac%7Bdr%7D%7Bdt%7D)
Implies that
velocity vector = ![(e^{t} cost - e^{t} sint)i + (e^{t} sint - e^{t}cost )j](https://tex.z-dn.net/?f=%28e%5E%7Bt%7D%20cost%20-%20e%5E%7Bt%7D%20sint%29i%20%2B%20%28e%5E%7Bt%7D%20sint%20-%20e%5E%7Bt%7Dcost%20%29j)
As acceleration is velocity over time so:
acceleration vector = a = ![\frac{dv}{dt}](https://tex.z-dn.net/?f=%5Cfrac%7Bdv%7D%7Bdt%7D)
Implies that
vector a =
![(e^{t}cost - e^{t}sint - e^{t}sint - e^{t}cost )i + ( e^{t}sint + e^{t}cost + e^{t}cost - e^{t}sint )j](https://tex.z-dn.net/?f=%28e%5E%7Bt%7Dcost%20-%20e%5E%7Bt%7Dsint%20-%20e%5E%7Bt%7Dsint%20-%20e%5E%7Bt%7Dcost%20%29i%20%2B%20%28%20e%5E%7Bt%7Dsint%20%2B%20e%5E%7Bt%7Dcost%20%2B%20e%5E%7Bt%7Dcost%20-%20e%5E%7Bt%7Dsint%20%29j)
vector a = ![(-2e^{t}sint) i + ( 2e^{t}cost)j](https://tex.z-dn.net/?f=%28-2e%5E%7Bt%7Dsint%29%20i%20%2B%20%28%202e%5E%7Bt%7Dcost%29j)
Now scalar product of position vector r and acceleration vector a:
r. a = ![. \\](https://tex.z-dn.net/?f=%3Ce%5E%7Bt%7Dcost%2C%20e%5E%7Bt%7Dsint%3E%20.%20%3C-2e%5E%7Bt%7Dsint%2C%202e%5E%7Bt%7Dcost%3E%5C%5C)
r.a = ![-2e^{2t}sintcost + 2e^{2t}sintcost](https://tex.z-dn.net/?f=-2e%5E%7B2t%7Dsintcost%20%2B%202e%5E%7B2t%7Dsintcost)
r.a = 0
Now, for angle between position vector r and acceleration vector a is given by:
cosФ =
= ![\frac{0}{|r|.|a|} = 0](https://tex.z-dn.net/?f=%5Cfrac%7B0%7D%7B%7Cr%7C.%7Ca%7C%7D%20%3D%200)
Ф =
(0) = 0
Hence, it is proved that angle between position vector r and acceleration vector a = 0 and is it never changes.
First, lets find what 6 percent of 36 is. We do this by multiplying 36 by 0.06.
36 * 0.06 = 2.16
After this, we add this amount to the total of 36 dollars, which will be the total of what she paid.
36 + 2.16 = 38.16
38.16 is the total of what she paid.
25-(2+5)(2+5)
25- 4+10+10+25
=66
Answer:
x = 40 degrees
Step-by-step explanation:
By corresponding angles, the angle above the 75 degree angle is also 75 degrees. Since angles of a triangle add to 180 degrees, x=180-75-65 = 40 degrees.