Question

Hello, could anyone please help me with my homework?

Solve for t.

`\mathrm{20=100(\displaystyle(1)/(2)) ^{\displaystyle(t)/(214) } }`
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Thank you in advance. :)

Answer 1

`\sf t= 496.9`

solving steps:

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

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

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

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

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

`\sf \bold{simplify}`

`\hookrightarrow \sf ln((1)/(5)) = (t)/(214) ln\:\left((1)/(2)\right)}`

`\hookrightarrow \sf (ln((1)/(5)) )/(ln((1)/(2))\right))} = (t)/(214)`

`\hookrightarrow \sf (214 \ ( \ ln((1)/(5)) \ ) )/(ln((1)/(2))\right))} = {t}`

`\hookrightarrow \sf t= 496.9`

Answer 2

Answer: See below

Explanation:

`100\left((1)/(2)\right)^{(t)/(214)}=20`

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

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

`Simplify`

`\left((1)/(2)\right)^{(t)/(214)}=(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((1)/(2)\right)^{(t)/(214)}\right)=\ln \left((1)/(5)\right)`

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

`\ln \left(\left((1)/(2)\right)^{(t)/(214)}\right)=(t)/(214)\ln \left((1)/(2)\right)`

`(t)/(214)\ln \left((1)/(2)\right)=\ln \left((1)/(5)\right)`

`(\ln \left((1)/(2)\right)t)/(214)=\ln \left((1)/(5)\right)`

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

`(214\ln \left((1)/(2)\right)t)/(214)=214\ln \left((1)/(5)\right)`

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

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

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

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