Logarithms: Solving equations 5

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From users who are members of Introduction to Calculus :

Wan Mekwi

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4 years, 6 months ago

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Wan Mekwi 4 years, 6 months ago

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Wan Mekwi 4 years, 6 months ago

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Wan Mekwi 4 years, 6 months ago

Created this as a copy of Logarithms: Solving equations 5.

Name	Status	Author	Last Modified
Logarithms: Solving equations 5	Ready to use	Bill Foster	13/10/2020 13:22
Logarithms: Solving equations 5	Ready to use	Wan Mekwi	13/10/2020 13:23

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Question statement

The size $\rho(t)$ of a population satisfies:

$\rho(t)=\rho_0e^{kt}$
where

$\rho$ and

$k$ are constants.

Given the following data, find $k$ in each case. Input your answers to 3 decimal places.

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	5
c	integer	3
b	integer	7
r1	integer	3
r2	integer	2
k	number	0.045
a1	integer	8
k1	number	0.145
r	integer	2
b1	integer	8
kvalue1	number	0.220
c1	integer	5
kvalue	number	0.231

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Description

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Generated value: `integer`

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Advice
"Part a)" - prompt

Parts

Part a) Gap-fill
1. Gap Number entry
  Add
Part b) Gap-fill
1. Gap Number entry
  Add

Gap-fill

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$\rho_0=\var{a} \times 10^{\var{b}},\;\;\;\;\;\rho(\var{c})=\var{a*r} \times 10^{\var{b}}$

$k=\;\;$ Unnamed gap

Advice

We have on solving the equation $\rho(t)=\rho_0e^{kt}$ by taking natural logs of both sides that:

$k = \frac{1}{t}\ln\left(\frac{\rho}{\rho_0}\right)$

$\begin{eqnarray*} k&=& \frac{1}{\var{c}}\ln\left(\frac{\var{a*r} \times 10^{\var{b}}}{\var{a} \times 10^{\var{b}}}\right)\\ &=& \frac{1}{\var{c}}\ln(\var{r})\\ &=&\var{kvalue} \end{eqnarray*}$ to 3 decimal places.

$\begin{eqnarray*} k&=& \frac{1}{\var{c1}}\ln\left(\frac{\var{a1*r1} \times 10^{\var{b1}}}{\var{a1} \times 10^{\var{b1}}}\right)\\ &=& \frac{1}{\var{c1}}\ln(\var{r1})\\ &=&\var{kvalue1} \end{eqnarray*}$ to 3 decimal places.

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Part	Test	Passed?
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Number entry
Number entry		Hasn't run yet
Gap-fill
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Number entry
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This question is used in the following exams:

Exponentials and Logarithms by Wan Mekwi in Introduction to Calculus.
Test Numbas Canvas by Joachim Latzer in Joachim's workspace.

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Wan Mekwi 4 years, 6 months ago

Wan Mekwi 4 years, 6 months ago

Wan Mekwi 4 years, 6 months ago

Error in variable testing condition

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Generated value: `integer`

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Logarithms: Solving equations 5

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Wan Mekwi 4 years, 6 months ago

Wan Mekwi 4 years, 6 months ago

Wan Mekwi 4 years, 6 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`