David's copy of Find bounds for distance and time spent running, given imprecise measurements

Error

There was an error loading the page.

Contributors

History

○

David Rickard 6 years, 5 months ago

Created this as a copy of Find bounds for distance and time spent running, given imprecise measurements.

Name	Status	Author	Last Modified
Limits of accuracy in measuring weight in a gym scenario	Ready to use	Stanislav Duris	20/11/2019 14:42
Find bounds for distance and time spent running, given imprecise measurements	Has some problems	Stanislav Duris	02/03/2020 10:31
David's copy of Find bounds for distance and time spent running, given imprecise measurements	draft	David Rickard	07/11/2018 21:06

There are 7 other versions that do you not have access to.

Question statement

{person['name']} is a keen runner. {capitalise(pronouns['they'])} run{verbs} at an average speed of {speed}km/h, rounded to the nearest integer.

Give any introductory information the student needs.

			Name	Type	Generated Value

speed	integer	7
distance	number	3.5
time	number	0.5
atime	integer	5
person	dict	Dictionary with 3 entries
pronouns	dict	Dictionary with 5 entries
verbs	string	s

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random(7..11)

JME function reference

Description

Speed in km/h.

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

Can an exam override the value of this variable?

Generated value: `integer`

This variable doesn't seem to be used anywhere.

Parts

Part a) Gap-fill
1. Gap Gap 0. Number entry
  Add
2. Gap Gap 1. Number entry
  Add
1. Step Information only
Part b) Gap-fill
1. Gap Gap 0. Number entry
  Add
2. Gap Gap 1. Number entry
  Add
Part c) Gap-fill

Gap-fill

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

Suppose {person['name']} ran for {atime} minutes, rounded to the nearest minute.

Using the rounded figures for {pronouns['their']} average speed and time spent running, calculate upper and lower bounds for the distance, $d$ , that {person['name']} ran.

Round your answers to two decimal places.

Gap 0. $\leq d \lt$ Gap 1.km

Advice

We're not certain about some of the measurements given in this question - we only know the rounded values. This means that the true value could be lower or higher than the given measurement.

We can find upper and lower bounds for the given measurements. Any values we go on to calculate will also be uncertain and have upper and lower bounds.

To find bounds for a given measurement, we divide the degree of accuracy by 2 and subtract or add this to our estimate to get lower and upper bounds respectively.

For example, $52$ rounded to the nearest integer has a lower bound of $51.5$ and an upper bound of $52.5$.

a)

The distance travelled is given by

\[ d = \text{Average speed} \times \text{Time taken} \]

We find bounds for speed and time first.

Lower bound for speed: $\var{speed} - 0.5 = \var{speed - 0.5} \text{ km/h}$

Upper bound for speed: $\var{speed} + 0.5 = \var{speed + 0.5} \text{ km/h}$

First, note that the speed is given in km/h and we want to find the distance in km. We will convert the given time into hours.

Lower bound for time taken:

\begin{align} \var{atime} - 0.5 &= \var{atime - 0.5} \text{ min} \\[0.5em]
&= \frac{\var{atime - 0.5}}{60} \text{ h}
\end{align}

Upper bound for time taken:

\begin{align} \var{atime} + 0.5 &= \var{atime + 0.5} \text{ min} \\[0.5em]
&= \frac{\var{atime + 0.5}}{60} \text{ h}
\end{align}

Since we're multiplying the speed and time together, the lower bound for distance is the slowest speed multiplied by the shortest time:

\begin{align}
\text{Lower bound} &= \text{lower bound for speed} \times \text{lower bound for time}\\
&= \var{speed - 0.5} \times \frac{\var{(atime - 0.5)}}{60} \\
&= \var{precround((speed-0.5)*(atime - 0.5)/60, 2 )} \text{ km} \quad \text{(rounded to 2 decimal places).}
\end{align}

The upper bound for distance is the fastest speed multiplied by the longest time:

\begin{align}
\text{Upper bound} &= \text{upper bound for speed} \times \text{upper bound for time} \\
&= \var{speed + 0.5} \times \frac{\var{atime + 0.5}}{60} \\
&= \var{precround((speed+0.5)*(atime + 0.5)/60, 2 )} \text{ km} \quad \text{(rounded to 2 decimal places).}
\end{align}

Hence,

\[\var{precround((speed-0.5)*(atime - 0.5)/60, 2 )} \leq d \lt \var{precround((speed+0.5)*(atime + 0.5)/60, 2 )} \text{.}\]

b)

We're told the speed and the distance travelled, so the time taken is given by

\[ t = \frac{\text{Distance travelled}}{\text{Average speed}} \]

We found upper and lower bounds for {person['name']}'s average speed above.

The distance of the evening run is given to the nearest kilometre, so we can compute bounds as follows:

Lower bound for distance: $\var{distance} - 0.5 = \var{distance - 0.5} \mathrm{km}$

Upper bound for distance: $\var{distance} + 0.5 = \var{distance + 0.5} \mathrm{km}$

The upper bound for the time taken is the longest distance divided by the slowest speed:

\begin{align}
\text{Upper bound} &= \text{upper bound for distance} \div \text{lower bound for speed} \\
&= \var{distance + 0.5} \div \var{speed - 0.5} \\
&= \var{(distance + 0.5)/(speed - 0.5)} \text{ hours.}
\end{align}

We're asked for the answer in minutes, to two decimal places.

\begin{align}
\var{(distance + 0.5)/(speed - 0.5)} \text{ hours} &= \var{(distance + 0.5)/(speed - 0.5)}\times 60 \text{ min} \\
&= \var{precround((distance + 0.5)/(speed - 0.5)*60, 2)} \text{ minutes} \quad \text{(rounded to 2 decimal places)}
\end{align}

The lower bound for time is the shortest distance divided by the fastest speed:

\begin{align}
\text{Lower bound} &= \text{lower bound for distance} \div \text{upper bound for speed} \\
&= \var{distance - 0.5} \div \var{speed + 0.5} \\
&= \var{(distance - 0.5)/(speed + 0.5)} \text{ hours.}
\end{align}

Converting into minutes, to two decimal places:

\begin{align}
\var{(distance - 0.5)/(speed + 0.5)} \text{ hours} &= \var{(distance - 0.5)/(speed + 0.5)}\times 60 \text{ min} \\
&= \var{precround((distance - 0.5)/(speed + 0.5)*60, 2)} \text{ min.}
\end{align}

Therefore, we cannot confidently say {person['name']}'s time was less than {time*60 +1} minutes as the upper bound for {pronouns['their']} time, $\var{precround((distance + 0.5)/(speed - 0.5)*60, 2)}$ minutes, is above this threshold.

Give a worked solution to the whole question.

Use this tab to check that this question works as expected.

There was an error which means the tests can't run:

Part	Test	Passed?
Gap-fill
Gap-fill		Hasn't run yet
Number entry
Number entry		Hasn't run yet
Number entry
Number entry		Hasn't run yet
Information only
Gap-fill
Gap-fill		Hasn't run yet
Number entry
Number entry		Hasn't run yet
Number entry
Number entry		Hasn't run yet
Gap-fill
Gap-fill		Hasn't run yet
Choose one from a list
Choose one from a list		Hasn't run yet
Choose one from a list
Choose one from a list		Hasn't run yet
Choose one from a list
Choose one from a list		Hasn't run yet

This question is not used in any exams.

Loading...

Error

David's copy of Find bounds for distance and time spent running, given imprecise measurements

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

David Rickard 6 years, 5 months ago

Error in variable testing condition

Error

Warning

Generated value: `integer`

Parts

Gap-fill

a)

b)

Loading...

Error

David's copy of Find bounds for distance and time spent running, given imprecise measurements

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

David Rickard 6 years, 5 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

a)

b)

Generated value: `integer`