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The volume of a cinder cone volcano depends on the radius (of the tip and the base) and the height of the volcano. This is expressed in the equation:

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### $V = \\frac{\\pi h}{3} ({R_1}^2+R_1r_2+{r_2}^2)$

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which can also be considered as the expanded form: $V = \\frac{\\pi h {R_1}^2}{3} + \\frac{\\pi h R_1r_2}{3} + \\frac{\\pi h {r_2}^2}{3}$

\n

where;
$V$ = volume ($km^2$)
$R_1$ = radius of the base of the volcano ($km$)
$r_2$ = radius of the volcano at a given height ($km$)
$h$ = height at given volume ($km$)

\n The above volcano has the following dimensions:

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• $R_1$ = $\\var{capr}km$
• \n
• $r_2$ = $\\var{r}km$
• \n
• $h$ = $\\var{h}km$
• \n
\n

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The notatation $\\frac{\\partial}{\\partial z}$ means the partial derivative with respect to $z$. We treat any other variables as constants and differentiate as normal with respect to $z$. For example, if $b = 7z + 2az^2 +csin(y)$, then $\\frac{\\partial b}{\\partial z} = 7 + 4az$.

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Note: To input the following into Numbas, type:

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
 R1 $\\rightarrow R_1$ R1*r2 $\\rightarrow R_1r_2$ r2 $\\rightarrow r_2$ pi*h $\\rightarrow \\pi h$ pi $\\rightarrow\\pi$ r2^2 $\\rightarrow {r_2}^2$
\n
\n
• You must type an asterisk (*) to indicate multiplication within expressions.
• \n
", "variables": {"capR": {"name": "capR", "description": "", "definition": "random(1,1.5,2,2.5,3,3.5,4)", "templateType": "anything", "group": "Ungrouped variables"}, "v": {"name": "v", "description": "", "definition": "((pi*h)/3)*(capr^2+(capr*r)+r^2)", "templateType": "anything", "group": "Ungrouped variables"}, "dvdcapr": {"name": "dvdcapr", "description": "", "definition": "siground(((pi*h)/3)*(2*capr+r),3)", "templateType": "anything", "group": "Ungrouped variables"}, "dvdr": {"name": "dvdr", "description": "", "definition": "siground(((pi*h)/3)*(2*r+capr),3)", "templateType": "anything", "group": "Ungrouped variables"}, "r": {"name": "r", "description": "", "definition": "capr-0.75", "templateType": "anything", "group": "Ungrouped variables"}, "dvdh": {"name": "dvdh", "description": "", "definition": "siground(((pi/3))*(capR^2+(capr*r)+r^2),3)", "templateType": "anything", "group": "Ungrouped variables"}, "h": {"name": "h", "description": "", "definition": "random(r+2.5)", "templateType": "anything", "group": "Ungrouped variables"}}, "parts": [{"scripts": {}, "showCorrectAnswer": true, "prompt": "

Work out an expression for the rate of change of volume with respect to height and hence work out the value of the rate of change for the above volcano:

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Expression: $\\frac{\\partial V}{\\partial h}$ = []

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Value: $\\frac{\\partial V}{\\partial h}$ = []

\n

\n

The first part of this question asks for a mathematical expression for the change in volume with respect to the change in height ($\\frac{\\partial V}{\\partial h}$).

\n
\n
• From the original equation for volume, differentiate with respect to $h$. It may help to see the other variables as constants that are just like numbers.
• \n
• In the equation $V = \\frac{\\pi h}{3} ({R_1}^2+R_1r_2+{r_2}^2)$, you can consider it to be $V = \\frac{\\pi h {R_1}^2}{3} + \\frac{\\pi h R_1r_2}{3} + \\frac{\\pi h {r_2}^2}{3}$ in its expanded form
• \n
• Now for the first term: $V = \\frac{\\pi h {R_1}^2}{3}$, you can substitute $\\pi \\times {R_1}^2$ for a constant, $a$, for example, to give $V = \\frac{a\\times h}{3}$. This may help you visualise the term more easily to differentiate it with respect to $h$
• \n
• The first term differentiates to $\\frac{a}{3}$ as you decrease the power of $h$ from 1 to 0, meaning $h$ is removed from the equation.
• \n
• Now try the same for the second and the third term to give an overall expression for $\\frac{\\partial V}{\\partial h}$ =
• \n
", "checkingaccuracy": 0.001, "expectedvariablenames": [], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst"}, {"scripts": {}, "showCorrectAnswer": true, "prompt": "

Once you have an expression for $\\frac{\\partial V}{\\partial h}$ and you have checked it is correct by submitting the above answer, substitute the given values in to obtain an overall value for the rate of change of volume with respect to height.

", "type": "information", "marks": 0, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst"}], "stepsPenalty": 0}, {"scripts": {}, "showCorrectAnswer": true, "prompt": "

Now work out an expression for the rate of change of volume with respect to radius of the base of the volcano, hence deduce the value for this rate of change:

\n

Expression: $\\frac{\\partial V}{\\partial R_1}$ =[]

\n

Value: $\\frac{\\partial V}{\\partial R_1}$ = []

\n

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The first part of this question asks for a mathematical expression for the change in volume with respect to the change in radius of the base ($R_1$): ($\\frac{\\partial V}{\\partial R_1}$).

\n
\n
• From the original equation for volume, differentiate with respect to $R_1$. It may help to see the other variables as constants that are just like numbers.
• \n
• In the equation $V = \\frac{\\pi h}{3} ({R_1}^2+R_1r_2+{r_2}^2)$, you can consider it to be $V = \\frac{\\pi h {R_1}^2}{3} + \\frac{\\pi h R_1r_2}{3} + \\frac{\\pi h {r_2}^2}{3}$ in its expanded form
• \n
• Now for the first term: $V = \\frac{\\pi h {R_1}^2}{3}$, you can substitute $\\pi \\times h$ for a constant, $a$, for example, to give $V = \\frac{a\\times {R_1}^2}{3}$. This may help you visualise the term more easily to differentiate it with respect to $R_1$
• \n
• The first term differentiates to $\\frac{2\\times a\\times R_1}{3}$ as you decrease the power of $R_1$ from 2 to 1, and multiply the numerator by the original power, 2.
• \n
• Now try the same for the second and the third term to give an overall expression for $\\frac{\\partial V}{\\partial R_1}$ =
• \n
", "expectedvariablenames": [], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst"}, {"scripts": {}, "showCorrectAnswer": true, "prompt": "

Once you have an expression for $\\frac{\\partial V}{\\partial R_1}$ and you have checked it is correct by submitting the above answer, substitute the given values in to obtain an overall value for the rate of change of volume with respect to radius of the base of the volcano.

", "type": "information", "marks": 0, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst"}], "stepsPenalty": 0}, {"scripts": {}, "showCorrectAnswer": true, "prompt": "

Work out an expression for the rate of change of volume with respect to radius of the tip of the volcano, r, hence work out the value of the rate of change for the above volcano:

\n

Expression: $\\frac{\\partial V}{\\partial r_2}$ = []

\n

Value: $\\frac{\\partial V}{\\partial r_2}$ =[]

\n

The first part of this question asks for a mathematical expression for the change in volume with respect to the change in radius of the tip ($r_2$): ($\\frac{\\partial V}{\\partial r_2}$).

\n
\n
• From the original equation for volume, differentiate with respect to $r_2$. It may help to see the other variables as constants that are just like numbers.
• \n
• In the equation $V = \\frac{\\pi h}{3} ({R_1}^2+R_1r_2+{r_2}^2)$, you can consider it to be $V = \\frac{\\pi h {R_1}^2}{3} + \\frac{\\pi h R_1r_2}{3} + \\frac{\\pi h {r_2}^2}{3}$ in its expanded form
• \n
• Now for the first term: $V = \\frac{\\pi h {R_1}^2}{3}$, you can substitute $\\pi \\times h\\times {R_1}^2$ for a constant, $a$, for example, to give $V = \\frac{a}{3}$. This may help you visualise the term more easily to differentiate it with respect to $r_2$
• \n
• The first term differentiates to 0 as you do not have $r_2$ in the term.
• \n
• Now try the same for the second and the third term to give an overall expression for $\\frac{\\partial V}{\\partial r_2}$ =
• \n
", "checkingaccuracy": 0.001, "expectedvariablenames": [], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst"}, {"scripts": {}, "showCorrectAnswer": true, "prompt": "

Once you have an expression for $\\frac{\\partial V}{\\partial r_2}$ and you have checked it is correct by submitting the above answer, substitute the given values in to obtain an overall value for the rate of change of volume with respect to radius of the tip of the volcano.

", "type": "information", "marks": 0, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst"}], "stepsPenalty": 0}], "tags": [], "extensions": [], "ungrouped_variables": ["capR", "r", "h", "v", "dvdh", "dvdcapr", "dvdr"], "variablesTest": {"maxRuns": 100, "condition": ""}, "rulesets": {}, "functions": {}, "variable_groups": [], "name": "5: Partial differentiation - Cone Volcano", "metadata": {"description": "

Apply partial diffenetiation in a geology problem.

The images below are for illustration only and show how the radius at height $h$ affects the overall volume. The graph for the equation to solve volume from radius at a given height is quadratic. It then becomes a linear graph when the first derivative is taken which shows the rate of change of volume with respect to radius. It then becomes a zero order graph when the second derivative is taken, where the rate of change of the rate of change is constant.

\n   \n

Part a)

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The first part of this question asks for a mathematical expression for the change in volume with respect to the change in height ($\\frac{\\partial V}{\\partial h}$).

\n
\n
• From the original equation for volume, differentiate with respect to $h$. It may help to see the other variables as constants that are just like numbers.
• \n
• In the equation $V = \\frac{\\pi h}{3} ({R_1}^2+R_1r_2+{r_2}^2)$, you can consider it to be $V = \\frac{\\pi h {R_1}^2}{3} + \\frac{\\pi h R_1r_2}{3} + \\frac{\\pi h {r_2}^2}{3}$ in its expanded form
• \n
• Now for the first term: $V = \\frac{\\pi h {R_1}^2}{3}$, you can substitute $\\pi \\times {R_1}^2$ for a constant, $a$, for example, to give $V = \\frac{a\\times h}{3}$. This may help you visualise the term more easily to differentiate it with respect to $h$
• \n
• The first term differentiates to $\\frac{a}{3}$ as you decrease the power of $h$ from 1 to 0, meaning $h$ is removed from the equation.
• \n
• The second term differentiates to $V = \\frac{b}{3}$ if we consider $\\pi\\times R_1\\times r_2$ to be the constant $b$, as you decrease the power of $h$ from 1 to 0, meaning $h$ is removed from the equation.
• \n
• The third term follows this similar pattern which differentiates to $V = \\frac{c}{3}$ if we consider $\\pi \\times {r_2}^2$ to be the constant $c$, as you decrease the power of $h$ from 1 to 0, meaning $h$ is removed from the equation.
• \n
• Overall, this gives:$\\frac{\\partial V}{\\partial h}$ = $\\frac{\\pi}{3}({R_1}^2+R_1r_2+{r_2}^2)$
• \n
\n

The second part of this question asks for you to deduce the value of this rate of change. This is done by substituting the values you are given in the question statement:

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• $\\frac{\\partial V}{\\partial h}$ = $\\frac{\\pi}{3}({\\var{capR}}^2+\\var{capR}\\times \\var{r}+\\var{r}^2)$
• \n
• = $\\var{dvdh}$ to 3 sig.fig.
• \n
\n

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Part b)

\n

The first part of this question asks for a mathematical expression for the change in volume with respect to the change in radius of the base ($R_1$): ($\\frac{\\partial V}{\\partial R_1}$).

\n
\n
• From the original equation for volume, differentiate with respect to $R_1$. It may help to see the other variables as constants that are just like numbers.
• \n
• In the equation $V = \\frac{\\pi h}{3} ({R_1}^2+R_1r_2+{r_2}^2)$, you can consider it to be $V = \\frac{\\pi h {R_1}^2}{3} + \\frac{\\pi h R_1r_2}{3} + \\frac{\\pi h {r_2}^2}{3}$ in its expanded form
• \n
• Now for the first term: $V = \\frac{\\pi h {R_1}^2}{3}$, you can substitute $\\pi \\times h$ for a constant, $a$, for example, to give $V = \\frac{a\\times {R_1}^2}{3}$. This may help you visualise the term more easily to differentiate it with respect to $R_1$
• \n
• The first term differentiates to $\\frac{2\\times a\\times R_1}{3}$ as you decrease the power of $R_1$ from 2 to 1, and multiply the numerator by the original power, 2.
• \n
• The second term differentiates to $\\frac{b}{3}$ if we consider $\\pi h r_2$ to be constant $b$, as you decrease the power of $R_1$ from 1 to 0, meaning $R_1$ is removed from the equation.
• \n
• The third term differntiates to nothing because you do not have $R_1$ in this term.
• \n
• Overall, this gives: $\\frac{\\partial V}{\\partial R_1}$ = $\\frac{\\pi\\times \\var{h}}{3}(2\\times \\var{capr}\\times \\var{r}+\\var{r})$
• \n
• = $\\var{dvdcapr}$ to 3 sig.fig.
• \n
\n

\n

\n

Part c)

\n

The first part of this question asks for a mathematical expression for the change in volume with respect to the change in radius of the tip ($r_2$): ($\\frac{\\partial V}{\\partial r_2}$).

\n
\n
• From the original equation for volume, differentiate with respect to $r_2$. It may help to see the other variables as constants that are just like numbers.
• \n
• In the equation $V = \\frac{\\pi h}{3} ({R_1}^2+R_1r_2+{r_2}^2)$, you can consider it to be $V = \\frac{\\pi h {R_1}^2}{3} + \\frac{\\pi h R_1r_2}{3} + \\frac{\\pi h {r_2}^2}{3}$ in its expanded form
• \n
• Now for the first term: $V = \\frac{\\pi h {R_1}^2}{3}$, you can substitute $\\pi \\times h\\times {R_1}^2$ for a constant, $a$, for example, to give $V = \\frac{a}{3}$. This may help you visualise the term more easily to differentiate it with respect to $r_2$
• \n
• The first term differentiates to 0 as you do not have $r_2$ in the term.
• \n
• The second term differentiates to $\\frac{b}{3}$ if we consider $\\pi h R_1$ to be constant $b$, as you decrease the power of $r_2$ from 1 to 0, meaning $r_2$ is removed from the equation.
• \n
• The third term differentiates to $\\frac{2\\times a\\times r_2}{3}$ as you decrease the power of $r_2$ from 2 to 1, and multiply the numerator by the original power, 2.
• \n
• Overall, this gives $\\frac{\\partial V}{\\partial r_2}$ = $\\frac{\\pi\\times h}{3}(2\\times \\var{r}+\\var{capr}$
• \n
• = $\\var{dvdr}$ to 3 sig.fig.
• \n
", "type": "question", "contributors": [{"name": "Sarah Turner", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/881/"}]}]}], "contributors": [{"name": "Sarah Turner", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/881/"}]}