Newton's law of cooling

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3 years, 1 month ago

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Martin Jones 3 years ago

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Christian Lawson-Perfect was given access to the Newton's law of cooling 3 years ago

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Martin Jones 3 years, 1 month ago

Created this as a copy of Newton's law of cooling (for exam).

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Question statement

For water in a particular boiler, the temperature $T$ , in $^\circ$ C, as it cools down, is given by Newton's Law of Cooling:

$T=\simplify{{Ta}+{T0-Ta}e^({-k}*t)}$

where $t$ is the time in seconds.

Give any introductory information the student needs.

			Name	Type	Generated Value

T0	integer	59
Ta	integer	16
k	number	0.004
t1	integer	780
Temp1	number	17.898758242
Temp2	integer	49
t2	number	66.1731385568
Temp1m	number	56.8211412742

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random(50..90)

JME function reference

Description

Describe what this variable represents, and list any assumptions made about its value.

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Generated value: `integer`

This variable doesn't seem to be used anywhere.

Parts

Part a) Gap-fill
1. Gap Number entry
  1. Alt Poor rounding
  2. Alt Used mins
  Add
Part b) Choose one from a list
Part c) Gap-fill
1. Gap Number entry
  1. Alt Poor rounding
  Add

Gap-fill

Ask the student a question, and give any hints about how they should answer this part.

Calculate the temperature after {t1/60} minutes, correct to 1 decimal place.

$T=$ Unnamed gap $^\circ$ C

Advice

Part a)

{t1/60} minutes is {t1} seconds, so we substitute $t=\var{t1}$ into the temperature formula:

$T=\var{Ta}+\var{T0-Ta}e^{\var{-k}\times\var{t1}}=\var{precround(Temp1,4)}$

which rounds to {precround(Temp1,1)} $^\circ$ C to 1 decimal place.

Part b)

Start by swapping sides:

$\simplify{{Ta}+{T0-Ta}e^({-k}*t)}=T$

Subtract $\var{Ta}$ :

$\simplify{{T0-Ta}e^({-k}*t)}=T-\var{Ta}$

Divide by $\var{T0-Ta}$ :

$\simplify{e^({-k}*t)}=\frac{T-\var{Ta}}{\var{T0-Ta}}$

Take natural log:

$\simplify{{-k}*t}=\ln\left(\frac{T-\var{Ta}}{\var{T0-Ta}}\right)$

And divide by $\var{-k}$ . A convenient way to do this, is to multiply by $-\frac{1}{\var{k}}$ :

$t=\simplify{-1/{k}*ln((T-{Ta})/({T0}-{Ta}))}$

Part c)

Substitute $T=\var{Temp2}$ into this formula:

$t=-\frac{1}{\var{k}}\times\ln\left(\frac{\var{Temp2}-\var{Ta}}{\var{T0-Ta}}\right)=\var{precround(t2,4)}$

This rounds to {precround(t2,0)} seconds.

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Choose one from a list
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This question is used in the following exams:

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HP48 outcomes 1&2 exam (Nov 2024) by Martin Jones in Martin's workspace.
HP48 outcomes 1&2 exam by Martin Jones in Martin's workspace.

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Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

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Checkpoint description

Martin Jones 3 years ago

Christian Lawson-Perfect was given access to the Newton's law of cooling 3 years ago

Martin Jones 3 years, 1 month ago

Martin Jones 3 years, 1 month ago

Error in variable testing condition

Error

Warning

Generated value: `integer`

Parts

Gap-fill

Part a)

Part b)

Part c)

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Newton's law of cooling

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Martin Jones 3 years ago

Christian Lawson-Perfect was given access to the Newton's law of cooling 3 years ago

Martin Jones 3 years, 1 month ago

Martin Jones 3 years, 1 month ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Part a)

Part b)

Part c)

Generated value: `integer`