The answer is x equals three
Answer:
<u>Mass</u>
![\sqrt{17}(\displaystyle\frac{8\pi^3}{3}+32\pi)](https://tex.z-dn.net/?f=%5Csqrt%7B17%7D%28%5Cdisplaystyle%5Cfrac%7B8%5Cpi%5E3%7D%7B3%7D%2B32%5Cpi%29)
<u>Center of mass</u>
<em>Coordinate x</em>
![\displaystyle\frac{(\displaystyle\frac{(2\pi)^4}{4}+32\pi)}{(\displaystyle\frac{8\pi^3}{3}+32\pi)}](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cfrac%7B%28%5Cdisplaystyle%5Cfrac%7B%282%5Cpi%29%5E4%7D%7B4%7D%2B32%5Cpi%29%7D%7B%28%5Cdisplaystyle%5Cfrac%7B8%5Cpi%5E3%7D%7B3%7D%2B32%5Cpi%29%7D)
<em>Coordinate y</em>
![\displaystyle\frac{16\pi}{(\displaystyle\frac{8\pi^3}{3}+32\pi)}](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cfrac%7B16%5Cpi%7D%7B%28%5Cdisplaystyle%5Cfrac%7B8%5Cpi%5E3%7D%7B3%7D%2B32%5Cpi%29%7D)
<em>Coordinate z</em>
![\displaystyle\frac{-16\pi}{(\displaystyle\frac{8\pi^3}{3}+32\pi)}](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cfrac%7B-16%5Cpi%7D%7B%28%5Cdisplaystyle%5Cfrac%7B8%5Cpi%5E3%7D%7B3%7D%2B32%5Cpi%29%7D)
Step-by-step explanation:
Let W be the wire. We can consider W=(x(t),y(t),z(t)) as a path given by the parametric functions
x(t) = t
y(t) = 4 cos(t)
z(t) = 4 sin(t)
for 0 ≤ t ≤ 2π
If D(x,y,z) is the density of W at a given point (x,y,z), the mass m would be the curve integral along the path W
![m=\displaystyle\int_{W}D(x,y,z)=\displaystyle\int_{0}^{2\pi}D(x(t),y(t),z(t))||W'(t)||dt](https://tex.z-dn.net/?f=m%3D%5Cdisplaystyle%5Cint_%7BW%7DD%28x%2Cy%2Cz%29%3D%5Cdisplaystyle%5Cint_%7B0%7D%5E%7B2%5Cpi%7DD%28x%28t%29%2Cy%28t%29%2Cz%28t%29%29%7C%7CW%27%28t%29%7C%7Cdt)
The density D(x,y,z) is given by
![D(x,y,z)=x^2+y^2+z^2=t^2+16cos^2(t)+16sin^2(t)=t^2+16](https://tex.z-dn.net/?f=D%28x%2Cy%2Cz%29%3Dx%5E2%2By%5E2%2Bz%5E2%3Dt%5E2%2B16cos%5E2%28t%29%2B16sin%5E2%28t%29%3Dt%5E2%2B16)
on the other hand
![||W'(t)||=\sqrt{1^2+(-4sin(t))^2+(4cos(t))^2}=\sqrt{1+16}=\sqrt{17}](https://tex.z-dn.net/?f=%7C%7CW%27%28t%29%7C%7C%3D%5Csqrt%7B1%5E2%2B%28-4sin%28t%29%29%5E2%2B%284cos%28t%29%29%5E2%7D%3D%5Csqrt%7B1%2B16%7D%3D%5Csqrt%7B17%7D)
and we have
![m=\displaystyle\int_{W}D(x,y,z)=\displaystyle\int_{0}^{2\pi}D(x(t),y(t),z(t))||W'(t)||dt=\\\\\sqrt{17}\displaystyle\int_{0}^{2\pi}(t^2+16)dt=\sqrt{17}(\displaystyle\frac{8\pi^3}{3}+32\pi)](https://tex.z-dn.net/?f=m%3D%5Cdisplaystyle%5Cint_%7BW%7DD%28x%2Cy%2Cz%29%3D%5Cdisplaystyle%5Cint_%7B0%7D%5E%7B2%5Cpi%7DD%28x%28t%29%2Cy%28t%29%2Cz%28t%29%29%7C%7CW%27%28t%29%7C%7Cdt%3D%5C%5C%5C%5C%5Csqrt%7B17%7D%5Cdisplaystyle%5Cint_%7B0%7D%5E%7B2%5Cpi%7D%28t%5E2%2B16%29dt%3D%5Csqrt%7B17%7D%28%5Cdisplaystyle%5Cfrac%7B8%5Cpi%5E3%7D%7B3%7D%2B32%5Cpi%29)
The center of mass is the point ![(\bar x,\bar y,\bar z)](https://tex.z-dn.net/?f=%28%5Cbar%20x%2C%5Cbar%20y%2C%5Cbar%20z%29)
where
![\bar x=\displaystyle\frac{1}{m}\displaystyle\int_{W}xD(x,y,z)\\\\\bar y=\displaystyle\frac{1}{m}\displaystyle\int_{W}yD(x,y,z)\\\\\bar z=\displaystyle\frac{1}{m}\displaystyle\int_{W}zD(x,y,z)](https://tex.z-dn.net/?f=%5Cbar%20x%3D%5Cdisplaystyle%5Cfrac%7B1%7D%7Bm%7D%5Cdisplaystyle%5Cint_%7BW%7DxD%28x%2Cy%2Cz%29%5C%5C%5C%5C%5Cbar%20y%3D%5Cdisplaystyle%5Cfrac%7B1%7D%7Bm%7D%5Cdisplaystyle%5Cint_%7BW%7DyD%28x%2Cy%2Cz%29%5C%5C%5C%5C%5Cbar%20z%3D%5Cdisplaystyle%5Cfrac%7B1%7D%7Bm%7D%5Cdisplaystyle%5Cint_%7BW%7DzD%28x%2Cy%2Cz%29)
We have
![\displaystyle\int_{W}xD(x,y,z)=\sqrt{17}\displaystyle\int_{0}^{2\pi}t(t^2+16)dt=\\\\=\sqrt{17}(\displaystyle\frac{(2\pi)^4}{4}+32\pi)](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint_%7BW%7DxD%28x%2Cy%2Cz%29%3D%5Csqrt%7B17%7D%5Cdisplaystyle%5Cint_%7B0%7D%5E%7B2%5Cpi%7Dt%28t%5E2%2B16%29dt%3D%5C%5C%5C%5C%3D%5Csqrt%7B17%7D%28%5Cdisplaystyle%5Cfrac%7B%282%5Cpi%29%5E4%7D%7B4%7D%2B32%5Cpi%29)
so
![\bar x=\displaystyle\frac{\sqrt{17}(\displaystyle\frac{(2\pi)^4}{4}+32\pi)}{\sqrt{17}(\displaystyle\frac{8\pi^3}{3}+32\pi)}=\displaystyle\frac{(\displaystyle\frac{(2\pi)^4}{4}+32\pi)}{(\displaystyle\frac{8\pi^3}{3}+32\pi)}](https://tex.z-dn.net/?f=%5Cbar%20x%3D%5Cdisplaystyle%5Cfrac%7B%5Csqrt%7B17%7D%28%5Cdisplaystyle%5Cfrac%7B%282%5Cpi%29%5E4%7D%7B4%7D%2B32%5Cpi%29%7D%7B%5Csqrt%7B17%7D%28%5Cdisplaystyle%5Cfrac%7B8%5Cpi%5E3%7D%7B3%7D%2B32%5Cpi%29%7D%3D%5Cdisplaystyle%5Cfrac%7B%28%5Cdisplaystyle%5Cfrac%7B%282%5Cpi%29%5E4%7D%7B4%7D%2B32%5Cpi%29%7D%7B%28%5Cdisplaystyle%5Cfrac%7B8%5Cpi%5E3%7D%7B3%7D%2B32%5Cpi%29%7D)
![\displaystyle\int_{W}yD(x,y,z)=\sqrt{17}\displaystyle\int_{0}^{2\pi}4cos(t)(t^2+16)dt=\\\\=16\sqrt{17}\pi](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint_%7BW%7DyD%28x%2Cy%2Cz%29%3D%5Csqrt%7B17%7D%5Cdisplaystyle%5Cint_%7B0%7D%5E%7B2%5Cpi%7D4cos%28t%29%28t%5E2%2B16%29dt%3D%5C%5C%5C%5C%3D16%5Csqrt%7B17%7D%5Cpi)
![\bar y=\displaystyle\frac{16\sqrt{17}\pi}{\sqrt{17}(\displaystyle\frac{8\pi^3}{3}+32\pi)}=\displaystyle\frac{16\pi}{(\displaystyle\frac{8\pi^3}{3}+32\pi)}](https://tex.z-dn.net/?f=%5Cbar%20y%3D%5Cdisplaystyle%5Cfrac%7B16%5Csqrt%7B17%7D%5Cpi%7D%7B%5Csqrt%7B17%7D%28%5Cdisplaystyle%5Cfrac%7B8%5Cpi%5E3%7D%7B3%7D%2B32%5Cpi%29%7D%3D%5Cdisplaystyle%5Cfrac%7B16%5Cpi%7D%7B%28%5Cdisplaystyle%5Cfrac%7B8%5Cpi%5E3%7D%7B3%7D%2B32%5Cpi%29%7D)
![\displaystyle\int_{W}zD(x,y,z)=4\sqrt{17}\displaystyle\int_{0}^{2\pi}sin(t)(t^2+16)dt=\\\\=-16\sqrt{17}\pi](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint_%7BW%7DzD%28x%2Cy%2Cz%29%3D4%5Csqrt%7B17%7D%5Cdisplaystyle%5Cint_%7B0%7D%5E%7B2%5Cpi%7Dsin%28t%29%28t%5E2%2B16%29dt%3D%5C%5C%5C%5C%3D-16%5Csqrt%7B17%7D%5Cpi)
![\bar z=\displaystyle\frac{-16\sqrt{17}\pi}{\sqrt{17}(\displaystyle\frac{8\pi^3}{3}+32\pi)}=\displaystyle\frac{-16\pi}{(\displaystyle\frac{8\pi^3}{3}+32\pi)}](https://tex.z-dn.net/?f=%5Cbar%20z%3D%5Cdisplaystyle%5Cfrac%7B-16%5Csqrt%7B17%7D%5Cpi%7D%7B%5Csqrt%7B17%7D%28%5Cdisplaystyle%5Cfrac%7B8%5Cpi%5E3%7D%7B3%7D%2B32%5Cpi%29%7D%3D%5Cdisplaystyle%5Cfrac%7B-16%5Cpi%7D%7B%28%5Cdisplaystyle%5Cfrac%7B8%5Cpi%5E3%7D%7B3%7D%2B32%5Cpi%29%7D)
Answer:
1). (x - 4 = 13)
2). (r + 17 = 51)
3). (7/9x =28)
4). (2/3x = 2)
5). { 3(x-2) = -9 }
6). (x -1.75 = 8.85)
7). Natalie buys organic almonds priced at $77 from the grocery store. How much did
she pay the cashier, if she received $23 in change?
Step-by-step explanation:
1. add 4 on both sides and it will be x=17
2. subtract 17 on both sides and it will be x= 34
3. multiply 9/7 on both sides and it will be x= 257/7 then you divide 257 by 7 then the final answer will be x= 36
4. multiply 3/2 on both sides and it will be x=6/2 then you divide 6 from 2 and the finial answer will be x= 3
5. you will distribute 3 into x and 2 and you will be 3x-6=-9 then you will add 6 on both sides so you will get 3x=3 and then divide 3 on both side so you will get x=1
6. add 1.75 on both sides so x= 10.6
7. the equation will be 23 +x = 77 so first you will subtract 23 on both sides and x will equal 54 so Natalie paid $54
The answer is 2... I think idrk 2 second ago was I thinking abt Damon Salvatore bunt GOOD LUCK!!!
you are correct. all angles (except central) are not equal (iff TU and CB are not parallel)
the triangles are not similar