Iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii dddddddddddddddoooooooooooooooonnnnnnnnnnnnnnn''''''''''''''ttttttttttttttttttttttttttttttttt kkkkkkkkkkkkkkkkkkkkkkkkkknnnnnnnnnnnnnnnnnnnnnnooooooooooooooowwwwwwwwwwwwwwwwwwwwwww
Answer:
A) -15, 5, 2
Step-by-step explanation:
2/3x - 14/3 = -2
3(2/3x - 14/3) = -2(3)
2x - 14 = -6
2x = -6 + 14
2x = 8
x = 8/2
x= 4
hope this helps :)
Answer:
26,508
Step-by-step explanation:
To find out how to solve this is that we first need to know that Triangular prisms have their own formula for finding surface area because they have two triangular faces opposite each other. The formula A=12bh is used to find the area of the top and bases triangular faces, where A = area, b = base, and h = height.
Also: To find the total surface area of a prism, you need to calculate the area of two polygonal bases, the top face and bottom face. And then calculate the area of lateral faces connecting the bases. Add up the area of the two bases and the area of the lateral faces to get the total surface area of a prism.
The top= 12 x 24 x 35 = 10,080
The lateral faces: 12 x 37 x 37 = 16,428
Surface Area= 10,080 + 16,428 = 26,508
We have to calculate the fourth roots of this complex number:
![z=9+9\sqrt[]{3}i](https://tex.z-dn.net/?f=z%3D9%2B9%5Csqrt%5B%5D%7B3%7Di)
We start by writing this number in exponential form:
![\begin{gathered} r=\sqrt[]{9^2+(9\sqrt[]{3})^2} \\ r=\sqrt[]{81+81\cdot3} \\ r=\sqrt[]{81+243} \\ r=\sqrt[]{324} \\ r=18 \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20r%3D%5Csqrt%5B%5D%7B9%5E2%2B%289%5Csqrt%5B%5D%7B3%7D%29%5E2%7D%20%5C%5C%20r%3D%5Csqrt%5B%5D%7B81%2B81%5Ccdot3%7D%20%5C%5C%20r%3D%5Csqrt%5B%5D%7B81%2B243%7D%20%5C%5C%20r%3D%5Csqrt%5B%5D%7B324%7D%20%5C%5C%20r%3D18%20%5Cend%7Bgathered%7D)
![\theta=\arctan (\frac{9\sqrt[]{3}}{9})=\arctan (\sqrt[]{3})=\frac{\pi}{3}](https://tex.z-dn.net/?f=%5Ctheta%3D%5Carctan%20%28%5Cfrac%7B9%5Csqrt%5B%5D%7B3%7D%7D%7B9%7D%29%3D%5Carctan%20%28%5Csqrt%5B%5D%7B3%7D%29%3D%5Cfrac%7B%5Cpi%7D%7B3%7D)
Then, the exponential form is:
The formula for the roots of a complex number can be written (in polar form) as:
Then, for a fourth root, we will have n = 4 and k = 0, 1, 2 and 3.
To simplify the calculations, we start by calculating the fourth root of r:
![r^{\frac{1}{4}}=18^{\frac{1}{4}}=\sqrt[4]{18}](https://tex.z-dn.net/?f=r%5E%7B%5Cfrac%7B1%7D%7B4%7D%7D%3D18%5E%7B%5Cfrac%7B1%7D%7B4%7D%7D%3D%5Csqrt%5B4%5D%7B18%7D)
<em>NOTE: It can not be simplified anymore, so we will leave it like this.</em>
Then, we calculate the arguments of the trigonometric functions:
We can now calculate for each value of k:
![\begin{gathered} k=0\colon \\ z_0=\sqrt[4]{18}\cdot(\cos (\pi(\frac{1}{8}+\frac{0}{2}))+i\cdot\sin (\pi(\frac{1}{8}+\frac{0}{2}))) \\ z_0=\sqrt[4]{18}\cdot(\cos (\frac{\pi}{8})+i\cdot\sin (\frac{\pi}{8}) \\ z_0=\sqrt[4]{18}\cdot e^{i\frac{\pi}{8}} \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20k%3D0%5Ccolon%20%5C%5C%20z_0%3D%5Csqrt%5B4%5D%7B18%7D%5Ccdot%28%5Ccos%20%28%5Cpi%28%5Cfrac%7B1%7D%7B8%7D%2B%5Cfrac%7B0%7D%7B2%7D%29%29%2Bi%5Ccdot%5Csin%20%28%5Cpi%28%5Cfrac%7B1%7D%7B8%7D%2B%5Cfrac%7B0%7D%7B2%7D%29%29%29%20%5C%5C%20z_0%3D%5Csqrt%5B4%5D%7B18%7D%5Ccdot%28%5Ccos%20%28%5Cfrac%7B%5Cpi%7D%7B8%7D%29%2Bi%5Ccdot%5Csin%20%28%5Cfrac%7B%5Cpi%7D%7B8%7D%29%20%5C%5C%20z_0%3D%5Csqrt%5B4%5D%7B18%7D%5Ccdot%20e%5E%7Bi%5Cfrac%7B%5Cpi%7D%7B8%7D%7D%20%5Cend%7Bgathered%7D)
![\begin{gathered} k=1\colon \\ z_1=\sqrt[4]{18}\cdot(\cos (\pi(\frac{1}{8}+\frac{1}{2}))+i\cdot\sin (\pi(\frac{1}{8}+\frac{1}{2}))) \\ z_1=\sqrt[4]{18}\cdot(\cos (\frac{5\pi}{8})+i\cdot\sin (\frac{5\pi}{8})) \\ z_1=\sqrt[4]{18}e^{i\frac{5\pi}{8}} \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20k%3D1%5Ccolon%20%5C%5C%20z_1%3D%5Csqrt%5B4%5D%7B18%7D%5Ccdot%28%5Ccos%20%28%5Cpi%28%5Cfrac%7B1%7D%7B8%7D%2B%5Cfrac%7B1%7D%7B2%7D%29%29%2Bi%5Ccdot%5Csin%20%28%5Cpi%28%5Cfrac%7B1%7D%7B8%7D%2B%5Cfrac%7B1%7D%7B2%7D%29%29%29%20%5C%5C%20z_1%3D%5Csqrt%5B4%5D%7B18%7D%5Ccdot%28%5Ccos%20%28%5Cfrac%7B5%5Cpi%7D%7B8%7D%29%2Bi%5Ccdot%5Csin%20%28%5Cfrac%7B5%5Cpi%7D%7B8%7D%29%29%20%5C%5C%20z_1%3D%5Csqrt%5B4%5D%7B18%7De%5E%7Bi%5Cfrac%7B5%5Cpi%7D%7B8%7D%7D%20%5Cend%7Bgathered%7D)
![\begin{gathered} k=2\colon \\ z_2=\sqrt[4]{18}\cdot(\cos (\pi(\frac{1}{8}+\frac{2}{2}))+i\cdot\sin (\pi(\frac{1}{8}+\frac{2}{2}))) \\ z_2=\sqrt[4]{18}\cdot(\cos (\frac{9\pi}{8})+i\cdot\sin (\frac{9\pi}{8})) \\ z_2=\sqrt[4]{18}e^{i\frac{9\pi}{8}} \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20k%3D2%5Ccolon%20%5C%5C%20z_2%3D%5Csqrt%5B4%5D%7B18%7D%5Ccdot%28%5Ccos%20%28%5Cpi%28%5Cfrac%7B1%7D%7B8%7D%2B%5Cfrac%7B2%7D%7B2%7D%29%29%2Bi%5Ccdot%5Csin%20%28%5Cpi%28%5Cfrac%7B1%7D%7B8%7D%2B%5Cfrac%7B2%7D%7B2%7D%29%29%29%20%5C%5C%20z_2%3D%5Csqrt%5B4%5D%7B18%7D%5Ccdot%28%5Ccos%20%28%5Cfrac%7B9%5Cpi%7D%7B8%7D%29%2Bi%5Ccdot%5Csin%20%28%5Cfrac%7B9%5Cpi%7D%7B8%7D%29%29%20%5C%5C%20z_2%3D%5Csqrt%5B4%5D%7B18%7De%5E%7Bi%5Cfrac%7B9%5Cpi%7D%7B8%7D%7D%20%5Cend%7Bgathered%7D)
![\begin{gathered} k=3\colon \\ z_3=\sqrt[4]{18}\cdot(\cos (\pi(\frac{1}{8}+\frac{3}{2}))+i\cdot\sin (\pi(\frac{1}{8}+\frac{3}{2}))) \\ z_3=\sqrt[4]{18}\cdot(\cos (\frac{13\pi}{8})+i\cdot\sin (\frac{13\pi}{8})) \\ z_3=\sqrt[4]{18}e^{i\frac{13\pi}{8}} \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20k%3D3%5Ccolon%20%5C%5C%20z_3%3D%5Csqrt%5B4%5D%7B18%7D%5Ccdot%28%5Ccos%20%28%5Cpi%28%5Cfrac%7B1%7D%7B8%7D%2B%5Cfrac%7B3%7D%7B2%7D%29%29%2Bi%5Ccdot%5Csin%20%28%5Cpi%28%5Cfrac%7B1%7D%7B8%7D%2B%5Cfrac%7B3%7D%7B2%7D%29%29%29%20%5C%5C%20z_3%3D%5Csqrt%5B4%5D%7B18%7D%5Ccdot%28%5Ccos%20%28%5Cfrac%7B13%5Cpi%7D%7B8%7D%29%2Bi%5Ccdot%5Csin%20%28%5Cfrac%7B13%5Cpi%7D%7B8%7D%29%29%20%5C%5C%20z_3%3D%5Csqrt%5B4%5D%7B18%7De%5E%7Bi%5Cfrac%7B13%5Cpi%7D%7B8%7D%7D%20%5Cend%7Bgathered%7D)
Answer:
The four roots in exponential form are
z0 = 18^(1/4)*e^(i*π/8)
z1 = 18^(1/4)*e^(i*5π/8)
z2 = 18^(1/4)*e^(i*9π/8)
z3 = 18^(1/4)*e^(i*13π/8)
