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 February 20th, 2012, 02:53 PM #1 Newbie   Joined: Nov 2011 Posts: 14 Thanks: 0 Word Problem The number of water hyacinth plants growing on a lake increases exponentially with time. On Tuesday the number of plants is 150. That Friday the number has risen to 240. If it takes 3400 plants to completely cover the lake, will the lake be completely covered by 20 days after Tuesday? Justify your answer.
February 20th, 2012, 04:12 PM   #2
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Re: Word Problem

Hello, doctorbleachers!

Quote:
 The number of water hyacinth plants growing on a lake increases exponentially with time. On Tuesday the number of plants is 150.[color=beige] .[/color]That Friday the number has risen to 240. If it takes 3400 plants to completely cover the lake, [color=beige]. . [/color]will the lake be completely covered by 20 days after Tuesday? Justify your answer.

$\text{The exponential function is: }\:P \;=\;P_oe^{kt}$

$\text{Tuesday: }\:t= 0,\;P = 150$

[color=beige]. . . . . . . . [/color]$150 \:=\:P_oe^0 \;\;\;\Rightarrow\;\;\;P_o = 150$

$\text{The function (so far) is: }\:P \:=\:150e^{kt}$

$\text{Friday: }\:t= 3,\;P = 240$

[color=beige]. . . . . . .[/color]$240 \:=\:150e^{3k} \;\;\;\Rightarrow\;\;\; e^{3k} \:=\:1.6 \;\;\;\Rightarrow\;\;\; \ln\left(e^{3k}\right) \:=\:\ln(1.6) \;\;\;\Rightarrow\;\;\;3k\underbrace{\ln(e)}_{\tex t{This is 1}} \:=\:\ln(1.6)$
[color=beige]. . . . . . . .[/color]$3k \:=\:\ln(1.6) \;\;\;\Rightarrow\;\;\;k \:=\:\frac{1}{3}\,\!\ln(1.6) \:=\:0.156667876 \;\approx\;0.157$

$\text{Therefore, the function is: }\:P \;=\;150e^{0.157t}$

$\text{When will {P= 3400\,?$

[color=beige]. . [/color]$3400 \:=\:150e^{0.157t} \;\;\;\Rightarrow\;\;\;e^{0.157t} \:=\:\frac{3400}{150} \:=\:\frac{68}{3} \;\;\;\Rightarrow\;\;\;\ln\left(e^{0.157t}\right) \:=\:\ln\left(\frac{68}{3}\right) \$

[color=beige]. . [/color]$0.157t \:=\:\ln\left(\frac{68}{3}\right) \;\;\;\Rightarrow\;\;\;t \:=\:\frac{\ln\left(\frac{68}{3}\right)}{0.157} \;=\; 19.87831475\text{ days.}$

$\text{The lake will be covered in about 19 days, 21 hours.}$

 February 20th, 2012, 05:18 PM #3 Global Moderator   Joined: Dec 2006 Posts: 20,926 Thanks: 2205 $P=150(1.6)^{t\small/3}.$ For t = 20, P = 3442.6 approximately, which exceeds 3400.
 February 20th, 2012, 08:33 PM #4 Newbie   Joined: Nov 2011 Posts: 14 Thanks: 0 Re: Word Problem Sweet. Thanks!!!

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### algebra 1 water hyacinths answer

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