Given:
![$\frac{\left(\frac{(4 r)^{3}}{15 t^{4}}\right)}{\left(\frac{16 r}{(3 t)^{2}}\right)}](https://tex.z-dn.net/?f=%24%5Cfrac%7B%5Cleft%28%5Cfrac%7B%284%20r%29%5E%7B3%7D%7D%7B15%20t%5E%7B4%7D%7D%5Cright%29%7D%7B%5Cleft%28%5Cfrac%7B16%20r%7D%7B%283%20t%29%5E%7B2%7D%7D%5Cright%29%7D)
To find:
The simplified fraction.
Solution:
Step 1: Simplify the numerator
![$\frac{(4 r)^{3}}{15 t^{4}}=\frac{4^3 r^{3}}{15 t^{4}}=\frac{64 r^{3}}{15 t^{4}}](https://tex.z-dn.net/?f=%24%5Cfrac%7B%284%20r%29%5E%7B3%7D%7D%7B15%20t%5E%7B4%7D%7D%3D%5Cfrac%7B4%5E3%20r%5E%7B3%7D%7D%7B15%20t%5E%7B4%7D%7D%3D%5Cfrac%7B64%20r%5E%7B3%7D%7D%7B15%20t%5E%7B4%7D%7D)
Step 2: Simplify the denominator
![$\frac{16 r}{(3 t)^{2}}=\frac{16 r}{3^2 t^{2}}= \frac{16 r}{9 t^{2}}](https://tex.z-dn.net/?f=%24%5Cfrac%7B16%20r%7D%7B%283%20t%29%5E%7B2%7D%7D%3D%5Cfrac%7B16%20r%7D%7B3%5E2%20t%5E%7B2%7D%7D%3D%20%5Cfrac%7B16%20r%7D%7B9%20t%5E%7B2%7D%7D)
Step 3: Using step 1 and step 2
![$\frac{\left(\frac{(4 r)^{3}}{15 t^{4}}\right)}{\left(\frac{16 r}{(3 t)^{2}}\right)}=\frac{\left(\frac{64 r^{3}}{15 t^{4}}\right)}{\left(\frac{16 r}{9 t^{2}} \right)}](https://tex.z-dn.net/?f=%24%5Cfrac%7B%5Cleft%28%5Cfrac%7B%284%20r%29%5E%7B3%7D%7D%7B15%20t%5E%7B4%7D%7D%5Cright%29%7D%7B%5Cleft%28%5Cfrac%7B16%20r%7D%7B%283%20t%29%5E%7B2%7D%7D%5Cright%29%7D%3D%5Cfrac%7B%5Cleft%28%5Cfrac%7B64%20r%5E%7B3%7D%7D%7B15%20t%5E%7B4%7D%7D%5Cright%29%7D%7B%5Cleft%28%5Cfrac%7B16%20r%7D%7B9%20t%5E%7B2%7D%7D%20%5Cright%29%7D)
Step 4: Using fraction rule:
![$\frac{\frac{a}{b}}{\frac{c}{d}}=\frac{a \cdot d}{b \cdot c}](https://tex.z-dn.net/?f=%24%5Cfrac%7B%5Cfrac%7Ba%7D%7Bb%7D%7D%7B%5Cfrac%7Bc%7D%7Bd%7D%7D%3D%5Cfrac%7Ba%20%5Ccdot%20d%7D%7Bb%20%5Ccdot%20c%7D)
![$\frac{\left(\frac{64 r^{3}}{15 t^{4}}\right)}{\left(\frac{16 r}{9 t^{2}}\right)}=\frac{64r^3 \cdot 9t^2}{16 r \cdot 15 t^4}](https://tex.z-dn.net/?f=%24%5Cfrac%7B%5Cleft%28%5Cfrac%7B64%20r%5E%7B3%7D%7D%7B15%20t%5E%7B4%7D%7D%5Cright%29%7D%7B%5Cleft%28%5Cfrac%7B16%20r%7D%7B9%20t%5E%7B2%7D%7D%5Cright%29%7D%3D%5Cfrac%7B64r%5E3%20%5Ccdot%209t%5E2%7D%7B16%20r%20%5Ccdot%2015%20t%5E4%7D)
Cancel the common factor r and t², we get
![$=\frac{64 r^{2} \cdot 9 }{16 \cdot 15 t^2 }](https://tex.z-dn.net/?f=%24%3D%5Cfrac%7B64%20r%5E%7B2%7D%20%5Ccdot%209%20%7D%7B16%20%20%5Ccdot%2015%20t%5E2%20%7D)
Cancel the common factors 16 and 3 on both numerator and denominator.
![$=\frac{4 r^{2} \cdot 3 }{ 5 t^2 }](https://tex.z-dn.net/?f=%24%3D%5Cfrac%7B4%20r%5E%7B2%7D%20%5Ccdot%203%20%7D%7B%20%205%20t%5E2%20%7D)
![$=\frac{12 r^{2} }{ 5 t^2 }](https://tex.z-dn.net/?f=%24%3D%5Cfrac%7B12%20r%5E%7B2%7D%20%20%7D%7B%20%205%20t%5E2%20%7D)
![$\frac{\left(\frac{(4 r)^{3}}{15 t^{4}}\right)}{\left(\frac{16 r}{(3 t)^{2}}\right)}=\frac{12 r^{2} }{ 5 t^2 }](https://tex.z-dn.net/?f=%24%5Cfrac%7B%5Cleft%28%5Cfrac%7B%284%20r%29%5E%7B3%7D%7D%7B15%20t%5E%7B4%7D%7D%5Cright%29%7D%7B%5Cleft%28%5Cfrac%7B16%20r%7D%7B%283%20t%29%5E%7B2%7D%7D%5Cright%29%7D%3D%5Cfrac%7B12%20r%5E%7B2%7D%20%20%7D%7B%20%205%20t%5E2%20%7D)
The simplified fraction is
.
Answer: 11.667
Step-by-step explanation:
The formula to find the test statistic for chi-square test is given by :-
![\chi^2=\dfrac{(n-1)\cdot s^2}{\sigma^2}](https://tex.z-dn.net/?f=%5Cchi%5E2%3D%5Cdfrac%7B%28n-1%29%5Ccdot%20s%5E2%7D%7B%5Csigma%5E2%7D)
Given : Claim : ![\sigma^2=12.6](https://tex.z-dn.net/?f=%5Csigma%5E2%3D12.6)
![n = 15\ ;\ s^2 = 10.5](https://tex.z-dn.net/?f=n%20%3D%2015%5C%20%3B%5C%20s%5E2%20%3D%2010.5)
Then , the standardized test statistic will be :-
![\chi^2=\dfrac{(15-1)\cdot 10.5}{12.6}\\\\=\dfrac{14\cdot 10.5}{12.6}=11.6666666667\approx11.667](https://tex.z-dn.net/?f=%5Cchi%5E2%3D%5Cdfrac%7B%2815-1%29%5Ccdot%2010.5%7D%7B12.6%7D%5C%5C%5C%5C%3D%5Cdfrac%7B14%5Ccdot%2010.5%7D%7B12.6%7D%3D11.6666666667%5Capprox11.667)
Hence, the standardized test statistic ![\chi^2=11.667](https://tex.z-dn.net/?f=%5Cchi%5E2%3D11.667)
Answer: (740)((1+0.0065)^132-1)/(0.0065)(1+0.0065)^132
Step-by-step explanation:
Answer:
.65
Step-by-step explanation:
searched it up