Question

Hello, could anyone please help me with my homework?

Answer 1

Answer: See below

Step-by-step explanation:

$100\left(\frac{1}{2}\right)^{\frac{t}{214}}=20$

$\mathrm{Divide\:both\:sides\:by\:}100$

$\frac{100\left(\frac{1}{2}\right)^{\frac{t}{214}}}{100}=\frac{20}{100}$

$Simplify$

$\left(\frac{1}{2}\right)^{\frac{t}{214}}=\frac{1}{5}$

$\mathrm{If\:}f\left(x\right)=g\left(x\right)\mathrm{,\:then\:}\ln \left(f\left(x\right)\right)=\ln \left(g\left(x\right)\right)$

$\ln \left(\left(\frac{1}{2}\right)^{\frac{t}{214}}\right)=\ln \left(\frac{1}{5}\right)$

$\mathrm{Apply\:log\:rule\:}\log _a\left(x^b\right)=b\cdot \log _a\left(x\right)$

$\ln \left(\left(\frac{1}{2}\right)^{\frac{t}{214}}\right)=\frac{t}{214}\ln \left(\frac{1}{2}\right)$

$\frac{t}{214}\ln \left(\frac{1}{2}\right)=\ln \left(\frac{1}{5}\right)$

$\frac{\ln \left(\frac{1}{2}\right)t}{214}=\ln \left(\frac{1}{5}\right)$

$\mathrm{Multiply\:both\:sides\:by\:}214$

$\frac{214\ln \left(\frac{1}{2}\right)t}{214}=214\ln \left(\frac{1}{5}\right)$

$-\ln \left(2\right)t=-214\ln \left(5\right)$

$\mathrm{Divide\:both\:sides\:by\:}-\ln \left(2\right)$

$\frac{-\ln \left(2\right)t}{-\ln \left(2\right)}=\frac{-214\ln \left(5\right)}{-\ln \left(2\right)}$

$t=\frac{214\ln \left(5\right)}{\ln \left(2\right)}$ or $\mathrm{Decimal}:\quad t=496.89261\dots$

Answer 2

Answer:

$\sf t= 496.9$

solving steps:

$\rightarrow \sf 20=\:100\left(\dfrac{1}{2}\right)^{\dfrac{t}{214}}$

$\sf \bold{divide \ both \ side \ by \ 100}$

$\hookrightarrow \sf \dfrac{1}{5} =\:\left(\dfrac{1}{2}\right)^{\dfrac{t}{214}}$

$\sf \bold{apply \ exponent \ rules}$

$\hookrightarrow \sf ln(\dfrac{1}{5}) = ln(\:\left(\dfrac{1}{2}\right)^{\dfrac{t}{214}})$

$\sf \bold{simplify}$

$\hookrightarrow \sf ln(\dfrac{1}{5}) = \dfrac{t}{214} ln\:\left(\dfrac{1}{2}\right)}$

$\hookrightarrow \sf \dfrac{ln(\dfrac{1}{5}) }{ln(\dfrac{1}{2})\right)}} = \dfrac{t}{214}$

$\hookrightarrow \sf \dfrac{214 \ ( \ ln(\dfrac{1}{5}) \ ) }{ln(\dfrac{1}{2})\right)}} = {t}$

$\hookrightarrow \sf t= 496.9$