Volume of revolution 2

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Bill Foster

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Bill Foster 55 years, 2 months ago

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Question statement

Consider the function

$y=f(x)=\var{a1}\ln(\var{m}x)$

Give any introductory information the student needs.

			Name	Type	Generated Value

vol	number	0.15
m	integer	6
a1	integer	3
tol	integer	0
td1	integer	2
tvol	number	0.1498078418
c1	integer	2
d1	integer	3

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
precround(tvol,2)

JME function reference

Description

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

Can an exam override the value of this variable?

Generated value: `number`

0.15

← Depends on:

tvol

This variable doesn't seem to be used anywhere.

Parts

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

Gap-fill

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

Rewrite this function in the form $x=g(y)$ , where $g(y)$ is a function of $y$ only.

$x=g(y)=\;\;$ Unnamed gap

Advice

a)
Given $y=\var{a1}\ln(\var{m}x)$ we rearrange so that $x$ is the subject of the equation:

$\begin{eqnarray*} y&=&\var{a1}\ln(\var{m}x)\\ \Rightarrow \ln(\var{m}x) &=&\frac{y}{\var{a1}}\\ \Rightarrow \var{m}x&=& e^{\frac{y}{\var{a1}}}\\ \Rightarrow x&=&\frac{1}{\var{m}}e^{\frac{y}{\var{a1}}} \end{eqnarray*}$
Hence

$g(y)=\frac{1}{\var{m}}e^{\frac{y}{\var{a1}}}$

b)
The volume of revolution is given by:

$V=\pi\int_{\var{c1}}^{\var{d1}}g(y)^2\;dy$
Using the expression for

$g(y)$ from the first part we have:

$\begin{eqnarray*} V&=&\pi\int_{\var{c1}}^{\var{d1}}\frac{1}{\var{m^2}}\simplify[std]{e^(2y/{a1})}\;dy\\ &=&\simplify[std]{({a1}/{2*m^2})*pi}\left[\simplify[std]{e^(2y/{a1})}\right]_{\var{c1}}^{\var{d1}}\\ \\&=&\var{tvol}\pi = \var{vol}\pi \end{eqnarray*}$
Hence the multiple of

$\pi$ to two decimal places is

$\var{vol}$ .

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Taxonomy: mathcentre

Taxonomy: Kind of activity

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Bill Foster 55 years, 2 months ago

Error in variable testing condition

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Generated value: `number`

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Volume of revolution 2

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Bill Foster 55 years, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: number

Parts

Gap-fill

Generated value: `number`