Math 266 Review Problems for Quiz Part 2 Answer Key Do the following problems as indicated.
6. The bacterium E. coli undergoes cell division approximately every 20 minutes. Starting with 100,000 cells, determine the number of cells after 2 hours.
Given: ! = !!!!"; !! = 100,000 !"##$; ! 20 !"#$%&' = 200,000 (Note: Cell division means that a cell divides itself to become 2 cells every 20 minutes.) Find: ! 2 ℎ!"#$ = !(120 !"#$%&') Solution: ! ! = 100,000!!" ! 20 = 200,000 ⇒ 100,000!!∙!" = 200,000 ⇒ !!∙!" = 2⟹ !!∙!"
!!" = 2
!!" ⟹
!! = 2!!" ⟹ ! ! = 100,000 ∙ 2
!!"!
! 120 = 100,000 ∙ 2!!"∙!"# = 100,000 ∙ 2! = 6,400,000
There will be 6,400,000 cells after 2 hours. 7. A physicist finds that an unknown radioactive substance registers 2000 counts per minute
on a Geiger counter. Ten days later the substance registers 1500 counts per minute. Approximate its half-‐life.
Given: ! = !!!!"; !! = 2000; ! 10 !"#$ = 1500 Find: Half-‐life T such that ! ! = !
!!!
Solution: ! ! = 2000!!"
! 10 = 1500 ⇒ 2000!!∙!" = 1500 ⇒ !!∙!" =15002000 =
34⟹ !!∙!"
!!" =
34
!!" ⟹
!! = 34
!!"⟹ ! ! = 2000 ∙
34
!!"!
! ! = 2000 ∙34
!!"!
= 1000 ⟹34
!!"!
=12 ⟹ ln
34
!!"!
= ln12 ⟹
!!"T ∙ ln !
!= ln !
! ⟹ ! = !" !".!
!".!"= 24 ⟹ Half life is 24 days; that is, it takes 24 days to half itself.
8. A metal plate that has been heated cools from 180℉ to 150℉ in 20 minutes when surrounded by air at a temperature of 60℉. Use Newton’s law of cooling to approximate its temperature at the end of one hour of cooling. When will the temperature be 100℉?
Given: ! ! = !! + !! − !! !!"; !! = 180°!; !! = 60°!;! 20 !"#$%&' = 150°! Find: ! 1 ℎ!"# = ! 60 !"#$%&' ; ! !ℎ!" ! ! = 100°! Solution: ! ! = 60+ 180− 60 !!" ⟹ ! ! = 60+ 120!!" ! 20 = 150 ⟹ 60+ 120!!∙!" = 150 ⟹ 120!!∙!" = 90 ⟹ !!∙!" = !"
!"#!.!" ⟹
!!∙!"!!" = . 75
!!" ⟹ ! ! = 60+ 120 . 75
!!"! ⟹
! 60 = 60+ 120 . 75!!"∙!" = 60+ 120 . 75 ! = 110.625°! Temperature after 1 hour
! ! = 100 ⟹ 60+ 120 . 75!!"! = 100 ⟹ 120 . 75
!!"! = 40 ⟹ . 75
!!"! = !"
!"#= !
! ⟹
ln . 75!!"! = ln !
! ⟹ !
!"T ∙ ln. 75 = ln !
! ⟹ ! = !" !" !!!".!"
= 76.4 minutes The temperature will be 100℉ at the end of 76.4 minute.
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