1 as they flipped a 6 sided cube and want to know the probability of rolling a 5
6-5=1
The answer is 1
Answer:
48
Step-by-step explanation:
80 over 100*150
80 divided by 100=0.8 *150=1208student
0.8 *60 = 48
Answer:
See below for all the cube roots
Step-by-step explanation:
<u>DeMoivre's Theorem</u>
Let 
be a complex number in polar form, where 
is an integer and 
. If 
, then 
.
<u>Nth Root of a Complex Number</u>
If is any positive integer, the nth roots of 
are given by ![\sqrt[n]{rcis\theta}=(rcis\theta)^{\frac{1}{n}}](https://tex.z-dn.net/?f=%5Csqrt%5Bn%5D%7Brcis%5Ctheta%7D%3D%28rcis%5Ctheta%29%5E%7B%5Cfrac%7B1%7D%7Bn%7D%7D)
where the nth roots are found with the formulas:
![\sqrt[n]{r}\biggr[cis(\frac{\theta+360^\circ k}{n})\biggr]](https://tex.z-dn.net/?f=%5Csqrt%5Bn%5D%7Br%7D%5Cbiggr%5Bcis%28%5Cfrac%7B%5Ctheta%2B360%5E%5Ccirc%20k%7D%7Bn%7D%29%5Cbiggr%5D)
for degrees (the one applicable to this problem)![\sqrt[n]{r}\biggr[cis(\frac{\theta+2\pi k}{n})\biggr]](https://tex.z-dn.net/?f=%5Csqrt%5Bn%5D%7Br%7D%5Cbiggr%5Bcis%28%5Cfrac%7B%5Ctheta%2B2%5Cpi%20k%7D%7Bn%7D%29%5Cbiggr%5D)
for radians
for 
<u>Calculation</u>
<u />![z=27(cos330^\circ+isin330^\circ)\\\\\sqrt[3]{z} =\sqrt[3]{27(cos330^\circ+isin330^\circ)}\\\\z^{\frac{1}{3}} =(27(cos330^\circ+isin330^\circ))^{\frac{1}{3}}\\\\z^{\frac{1}{3}} =27^{\frac{1}{3}}(cos(\frac{1}{3}\cdot330^\circ)+isin(\frac{1}{3}\cdot330^\circ))\\\\z^{\frac{1}{3}} =3(cos110^\circ+isin110^\circ)](https://tex.z-dn.net/?f=z%3D27%28cos330%5E%5Ccirc%2Bisin330%5E%5Ccirc%29%5C%5C%5C%5C%5Csqrt%5B3%5D%7Bz%7D%20%3D%5Csqrt%5B3%5D%7B27%28cos330%5E%5Ccirc%2Bisin330%5E%5Ccirc%29%7D%5C%5C%5C%5Cz%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D%20%3D%2827%28cos330%5E%5Ccirc%2Bisin330%5E%5Ccirc%29%29%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D%5C%5C%5C%5Cz%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D%20%3D27%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D%28cos%28%5Cfrac%7B1%7D%7B3%7D%5Ccdot330%5E%5Ccirc%29%2Bisin%28%5Cfrac%7B1%7D%7B3%7D%5Ccdot330%5E%5Ccirc%29%29%5C%5C%5C%5Cz%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D%20%3D3%28cos110%5E%5Ccirc%2Bisin110%5E%5Ccirc%29)
<u>First cube root where k=2</u>
<u />![\sqrt[3]{27}\biggr[cis(\frac{330^\circ+360^\circ(2)}{3})\biggr]\\3\biggr[cis(\frac{330^\circ+720^\circ}{3})\biggr]\\3\biggr[cis(\frac{1050^\circ}{3})\biggr]\\3\biggr[cis(350^\circ)\biggr]\\3\biggr[cos(350^\circ)+isin(350^\circ)\biggr]](https://tex.z-dn.net/?f=%5Csqrt%5B3%5D%7B27%7D%5Cbiggr%5Bcis%28%5Cfrac%7B330%5E%5Ccirc%2B360%5E%5Ccirc%282%29%7D%7B3%7D%29%5Cbiggr%5D%5C%5C3%5Cbiggr%5Bcis%28%5Cfrac%7B330%5E%5Ccirc%2B720%5E%5Ccirc%7D%7B3%7D%29%5Cbiggr%5D%5C%5C3%5Cbiggr%5Bcis%28%5Cfrac%7B1050%5E%5Ccirc%7D%7B3%7D%29%5Cbiggr%5D%5C%5C3%5Cbiggr%5Bcis%28350%5E%5Ccirc%29%5Cbiggr%5D%5C%5C3%5Cbiggr%5Bcos%28350%5E%5Ccirc%29%2Bisin%28350%5E%5Ccirc%29%5Cbiggr%5D)
<u>Second cube root where k=1</u>
![\sqrt[3]{27}\biggr[cis(\frac{330^\circ+360^\circ(1)}{3})\biggr]\\3\biggr[cis(\frac{330^\circ+360^\circ}{3})\biggr]\\3\biggr[cis(\frac{690^\circ}{3})\biggr]\\3\biggr[cis(230^\circ)\biggr]\\3\biggr[cos(230^\circ)+isin(230^\circ)\biggr]](https://tex.z-dn.net/?f=%5Csqrt%5B3%5D%7B27%7D%5Cbiggr%5Bcis%28%5Cfrac%7B330%5E%5Ccirc%2B360%5E%5Ccirc%281%29%7D%7B3%7D%29%5Cbiggr%5D%5C%5C3%5Cbiggr%5Bcis%28%5Cfrac%7B330%5E%5Ccirc%2B360%5E%5Ccirc%7D%7B3%7D%29%5Cbiggr%5D%5C%5C3%5Cbiggr%5Bcis%28%5Cfrac%7B690%5E%5Ccirc%7D%7B3%7D%29%5Cbiggr%5D%5C%5C3%5Cbiggr%5Bcis%28230%5E%5Ccirc%29%5Cbiggr%5D%5C%5C3%5Cbiggr%5Bcos%28230%5E%5Ccirc%29%2Bisin%28230%5E%5Ccirc%29%5Cbiggr%5D)
<u>Third cube root where k=0</u>
<u />![\sqrt[3]{27}\biggr[cis(\frac{330^\circ+360^\circ(0)}{3})\biggr]\\3\biggr[cis(\frac{330^\circ}{3})\biggr]\\3\biggr[cis(110^\circ)\biggr]\\3\biggr[cos(110^\circ)+isin(110^\circ)\biggr]](https://tex.z-dn.net/?f=%5Csqrt%5B3%5D%7B27%7D%5Cbiggr%5Bcis%28%5Cfrac%7B330%5E%5Ccirc%2B360%5E%5Ccirc%280%29%7D%7B3%7D%29%5Cbiggr%5D%5C%5C3%5Cbiggr%5Bcis%28%5Cfrac%7B330%5E%5Ccirc%7D%7B3%7D%29%5Cbiggr%5D%5C%5C3%5Cbiggr%5Bcis%28110%5E%5Ccirc%29%5Cbiggr%5D%5C%5C3%5Cbiggr%5Bcos%28110%5E%5Ccirc%29%2Bisin%28110%5E%5Ccirc%29%5Cbiggr%5D)
Answer:
P = dependant; t = independant
Step-by-step explanation:
The t value can be changed, making it the indepentant. The value of P changes as the value of t is changed, therefore making it dependant on t, making it the dependant value.
