can any one help me solve this 4000exp^(-400t)-50000exp^(-1000t)-64000exp^(-1600t)=0
G Guest Guest Jun 17, 2006 #1 can any one help me solve this 4000exp^(-400t)-50000exp^(-1000t)-64000exp^(-1600t)=0
D daon Senior Member Joined Jan 27, 2006 Messages 1,284 Jun 17, 2006 #2 4000exp^(-400t)-50000exp^(-1000t)-64000exp^(-1600t)=0 ... 50000e−1000t=4000e−400t−64000e−1600t\displaystyle 50000e^{-1000t} = 4000e^{-400t} - 64000e^{-1600t}50000e−1000t=4000e−400t−64000e−1600t Let u=e−200t\displaystyle u = e^{-200t}u=e−200t Then: 50000u5=4000u2−64000u8\displaystyle 50000u^5 = 4000u^2 - 64000u^850000u5=4000u2−64000u8 u2(−64000u6+50000u3+4000)=0\displaystyle u^2(-64000u^6 + 50000u^3 + 4000) = 0u2(−64000u6+50000u3+4000)=0 Let x=u3\displaystyle x = u^3x=u3 Then: -1000u2(64x2−50x−4)=0\displaystyle 1000u^2(64x^2 - 50x - 4) = 01000u2(64x2−50x−4)=0 Then set -1000u^2 = 0 and 64x^2 - 50x - 4 = 0 Use quadratic equation from here...
4000exp^(-400t)-50000exp^(-1000t)-64000exp^(-1600t)=0 ... 50000e−1000t=4000e−400t−64000e−1600t\displaystyle 50000e^{-1000t} = 4000e^{-400t} - 64000e^{-1600t}50000e−1000t=4000e−400t−64000e−1600t Let u=e−200t\displaystyle u = e^{-200t}u=e−200t Then: 50000u5=4000u2−64000u8\displaystyle 50000u^5 = 4000u^2 - 64000u^850000u5=4000u2−64000u8 u2(−64000u6+50000u3+4000)=0\displaystyle u^2(-64000u^6 + 50000u^3 + 4000) = 0u2(−64000u6+50000u3+4000)=0 Let x=u3\displaystyle x = u^3x=u3 Then: -1000u2(64x2−50x−4)=0\displaystyle 1000u^2(64x^2 - 50x - 4) = 01000u2(64x2−50x−4)=0 Then set -1000u^2 = 0 and 64x^2 - 50x - 4 = 0 Use quadratic equation from here...
S soroban Elite Member Joined Jan 28, 2005 Messages 5,586 Jun 18, 2006 #3 Hello, billybob_bonka! My approach varies slightly . . . Can any one help me solve this? 4,000e−400t − 50,000e−1000t − 64,000e−1600t = 0\displaystyle 4,000e^{^{-400t}}\,-\,50,000e^{^{-1000t}}\,-\,64,000e^{^{-1600t}}\;=\;04,000e−400t−50,000e−1000t−64,000e−1600t=0 Click to expand... Divide by 2,000 ⋯ 2e400t − 25e−1000t − 32e−1600t = 0\displaystyle 2,000\;\cdots\;2e^{^{400t}}\,-\,25e^{^{-1000t}}\,-\,32e^{^{-1600t}}\;=\;02,000⋯2e400t−25e−1000t−32e−1600t=0 Multiply by e1600t ⋯ 2e1200t − 25e600t − 32 = 0\displaystyle e^{^{1600t}}\;\cdots\;2e^{^{1200t}}\,-\,25e^{^{600t}}\,-\,32\;=\;0e1600t⋯2e1200t−25e600t−32=0 We have a quadratic: 2(e600t)2 − 25e600t − 32 = 0\displaystyle \,2(e^{^{600t}})^2\,-\,25e^{^{600t}}\,-\,32\;=\;02(e600t)2−25e600t−32=0 Let u = e600t ⋯ 2u2 − 25u − 32 = 0\displaystyle u\,=\,e^{^{600t}}\;\cdots\;2u^2\,-\,25u\,-\,32\;=\;0u=e600t⋯2u2−25u−32=0 Quadratic Formula: u = 25 ± 252 − 4(2)(−32)2(2) = 25 ± 8814\displaystyle \:u\;=\;\frac{25\,\pm\,\sqrt{25^2\,-\,4(2)(-32)}}{2(2)} \;= \;\frac{25\,\pm\,\sqrt{881}}{4}u=2(2)25±252−4(2)(−32)=425±881 Then we have: e600t = 25 + 8814 ⇒ 600t = ln(25 + 8814)\displaystyle \,e^{^{600t}}\:=\;\frac{25\,+\,\sqrt{881}}{4}\;\;\Rightarrow\;\;600t\:=\:\ln\left(\frac{25\,+\,\sqrt{881}}{4}\right)e600t=425+881⇒600t=ln(425+881) Therefore: t = 1600ln(25 + 8814) = 0.004358723...\displaystyle \:t\;=\;\frac{1}{600}\ln\left(\frac{25\,+\,\sqrt{881}}{4}\right)\;=\;0.004358723...t=6001ln(425+881)=0.004358723...
Hello, billybob_bonka! My approach varies slightly . . . Can any one help me solve this? 4,000e−400t − 50,000e−1000t − 64,000e−1600t = 0\displaystyle 4,000e^{^{-400t}}\,-\,50,000e^{^{-1000t}}\,-\,64,000e^{^{-1600t}}\;=\;04,000e−400t−50,000e−1000t−64,000e−1600t=0 Click to expand... Divide by 2,000 ⋯ 2e400t − 25e−1000t − 32e−1600t = 0\displaystyle 2,000\;\cdots\;2e^{^{400t}}\,-\,25e^{^{-1000t}}\,-\,32e^{^{-1600t}}\;=\;02,000⋯2e400t−25e−1000t−32e−1600t=0 Multiply by e1600t ⋯ 2e1200t − 25e600t − 32 = 0\displaystyle e^{^{1600t}}\;\cdots\;2e^{^{1200t}}\,-\,25e^{^{600t}}\,-\,32\;=\;0e1600t⋯2e1200t−25e600t−32=0 We have a quadratic: 2(e600t)2 − 25e600t − 32 = 0\displaystyle \,2(e^{^{600t}})^2\,-\,25e^{^{600t}}\,-\,32\;=\;02(e600t)2−25e600t−32=0 Let u = e600t ⋯ 2u2 − 25u − 32 = 0\displaystyle u\,=\,e^{^{600t}}\;\cdots\;2u^2\,-\,25u\,-\,32\;=\;0u=e600t⋯2u2−25u−32=0 Quadratic Formula: u = 25 ± 252 − 4(2)(−32)2(2) = 25 ± 8814\displaystyle \:u\;=\;\frac{25\,\pm\,\sqrt{25^2\,-\,4(2)(-32)}}{2(2)} \;= \;\frac{25\,\pm\,\sqrt{881}}{4}u=2(2)25±252−4(2)(−32)=425±881 Then we have: e600t = 25 + 8814 ⇒ 600t = ln(25 + 8814)\displaystyle \,e^{^{600t}}\:=\;\frac{25\,+\,\sqrt{881}}{4}\;\;\Rightarrow\;\;600t\:=\:\ln\left(\frac{25\,+\,\sqrt{881}}{4}\right)e600t=425+881⇒600t=ln(425+881) Therefore: t = 1600ln(25 + 8814) = 0.004358723...\displaystyle \:t\;=\;\frac{1}{600}\ln\left(\frac{25\,+\,\sqrt{881}}{4}\right)\;=\;0.004358723...t=6001ln(425+881)=0.004358723...
