Answer:
![\mathbf{v}(\pi/4)=-3\mathbf{i}+3\mathbf{j}+3 \mathbf{k}](https://tex.z-dn.net/?f=%5Cmathbf%7Bv%7D%28%5Cpi%2F4%29%3D-3%5Cmathbf%7Bi%7D%2B3%5Cmathbf%7Bj%7D%2B3%20%5Cmathbf%7Bk%7D)
Step-by-step explanation:
<u>Instantaneous Velocity</u>
Given r(t) as the vector function of the position for time t, the instantaneous velocity is computed as:
![\mathbf{v}=\frac{d\mathbf{r}}{dt}=\mathbf{r}'(t)](https://tex.z-dn.net/?f=%5Cmathbf%7Bv%7D%3D%5Cfrac%7Bd%5Cmathbf%7Br%7D%7D%7Bdt%7D%3D%5Cmathbf%7Br%7D%27%28t%29)
We are given:
![\mathbf{r}=\sin^2(3t)\mathbf{i}+3t\mathbf{j}-\cos^2(3t)\mathbf{k}](https://tex.z-dn.net/?f=%5Cmathbf%7Br%7D%3D%5Csin%5E2%283t%29%5Cmathbf%7Bi%7D%2B3t%5Cmathbf%7Bj%7D-%5Ccos%5E2%283t%29%5Cmathbf%7Bk%7D)
Thus:
![\mathbf{v}(t)=\mathbf{r}'(t)=[\sin^2(3t)\mathbf{i}+3t\mathbf{j}-\cos^2(3t)\mathbf{k}]'](https://tex.z-dn.net/?f=%5Cmathbf%7Bv%7D%28t%29%3D%5Cmathbf%7Br%7D%27%28t%29%3D%5B%5Csin%5E2%283t%29%5Cmathbf%7Bi%7D%2B3t%5Cmathbf%7Bj%7D-%5Ccos%5E2%283t%29%5Cmathbf%7Bk%7D%5D%27)
Computing the derivative:
![\mathbf{v}(t)=2\sin(3t)\cos(3t)(3)\mathbf{i}+3\mathbf{j}+2\cos(3t)\sin(3t)(3)\mathbf{k}](https://tex.z-dn.net/?f=%5Cmathbf%7Bv%7D%28t%29%3D2%5Csin%283t%29%5Ccos%283t%29%283%29%5Cmathbf%7Bi%7D%2B3%5Cmathbf%7Bj%7D%2B2%5Ccos%283t%29%5Csin%283t%29%283%29%5Cmathbf%7Bk%7D)
![\mathbf{v}(t)=6\sin(3t)\cos(3t)\mathbf{i}+3\mathbf{j}+6\cos(3t)\sin(3t)\mathbf{k}](https://tex.z-dn.net/?f=%5Cmathbf%7Bv%7D%28t%29%3D6%5Csin%283t%29%5Ccos%283t%29%5Cmathbf%7Bi%7D%2B3%5Cmathbf%7Bj%7D%2B6%5Ccos%283t%29%5Csin%283t%29%5Cmathbf%7Bk%7D)
Evaluating for t=π/4:
![\mathbf{v}(\pi/4)=6\sin(3\pi/4)\cos(3\pi/4)\mathbf{i}+3\mathbf{j}+6\cos(3\pi/4)\sin(3\pi/4) \mathbf{k}](https://tex.z-dn.net/?f=%5Cmathbf%7Bv%7D%28%5Cpi%2F4%29%3D6%5Csin%283%5Cpi%2F4%29%5Ccos%283%5Cpi%2F4%29%5Cmathbf%7Bi%7D%2B3%5Cmathbf%7Bj%7D%2B6%5Ccos%283%5Cpi%2F4%29%5Csin%283%5Cpi%2F4%29%20%5Cmathbf%7Bk%7D)
![\mathbf{v}(\pi/4)=6(\sqrt{2}/2)(-\sqrt{2}/2)\mathbf{i}+3\mathbf{j}+6(-\sqrt{2}/2))(\sqrt{2}/2)) \mathbf{k}](https://tex.z-dn.net/?f=%5Cmathbf%7Bv%7D%28%5Cpi%2F4%29%3D6%28%5Csqrt%7B2%7D%2F2%29%28-%5Csqrt%7B2%7D%2F2%29%5Cmathbf%7Bi%7D%2B3%5Cmathbf%7Bj%7D%2B6%28-%5Csqrt%7B2%7D%2F2%29%29%28%5Csqrt%7B2%7D%2F2%29%29%20%5Cmathbf%7Bk%7D)
![\mathbf{v}(\pi/4)=-3\mathbf{i}+3\mathbf{j}-3 \mathbf{k}](https://tex.z-dn.net/?f=%5Cmathbf%7Bv%7D%28%5Cpi%2F4%29%3D-3%5Cmathbf%7Bi%7D%2B3%5Cmathbf%7Bj%7D-3%20%5Cmathbf%7Bk%7D)
Answer:
52
Step-by-step explanation:
your mother is not okay
A common denominator would be 24
Answer:
x = -y - 4 or x = - (y + 4)
Step-by-step explanation:
p(x) = -x - 4
y = -x - 4
y + x = -x +x - 4
y + x = -4
y - y +x = -y - 4
x = - (y + 4)
Answer:
A. 56-6m<20
B. 2.95b+28<450
13. Less than, greater than, going over or not going over.
Step-by-step explanation:
A. The answer to the first question can be simple. If she can scan 6 photographs per minute, then we would write the inequality 56-6m<20, since we are subtracting 6 from 56 every minute, we would write 6 per minute as 6m, and subtract that from 56. Lastly, since it says <u>less than</u>, we need to put the less than symbol, the mouth facing towards the 20 as the greater number.
B. If she has spent 28, that means that we put -28 on the equation as a constant. Then, we put the price times the number of batteries she bought, or 2.95b, and put a less than symbol, since we want to spend less than the 450 gift card. So the equation would be 2.95b+28<450
13. Key words to indicate inequalities can be less than, greater than, without going over, going over.
Hope this helps! :D