// Numbas version: exam_results_page_options {"name": "Differentiation 18 - Applications", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["v", "a"], "name": "Differentiation 18 - Applications", "tags": [], "advice": "

This section draws on the skills learnt the previous parts of the 'Differentiation' series of questions, and some geometry knowledge.

\n

The hint under the steps should be all the extra information you need.

", "rulesets": {}, "parts": [{"prompt": "

The height of the tank

\n

$h=$ [[0]] m

\n

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": ["x"], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "{v}/x^2", "marks": "2", "checkvariablenames": true, "checkingtype": "absdiff", "vsetrange": [0, 1]}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "

The surface area of the tank

\n

$S=$ [[0]] $\\text{m}^2$

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": ["x"], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "x^2+4{v}/x", "marks": "2", "checkvariablenames": true, "checkingtype": "absdiff", "vsetrange": [0, 1]}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"stepsPenalty": "1", "prompt": "

Given that the surface area is a minimum, find the value of $x$.

\n

$S'(x)=$ [[1]]

\n

Therefore,

\n

$x=$ [[0]]     (give your answer to 2 decimal places)

\n

Check this is a minimum.

\n

$S''(x)=$ [[2]]

\n

Substitute your value for $x$ into $S''(x)$ and determine whether is it a minimum.

\n

Type '$Y$' for yes, '$N$' for no, or '$U$' for undefined.

\n

[[3]]

\n

\n

Hence, the minimum area of metal used is

\n

$A_{min}=$ [[4]]     (give your answer to 2 decimal places)

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "precision": "2", "maxValue": "{a}", "minValue": "{a}", "variableReplacementStrategy": "originalfirst", "strictPrecision": false, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": "50", "scripts": {}, "marks": "2", "type": "numberentry", "showPrecisionHint": false}, {"vsetrangepoints": 5, "expectedvariablenames": ["x"], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "2x-4{v}/x^2", "marks": 1, "checkvariablenames": true, "checkingtype": "absdiff", "vsetrange": [0, 1]}, {"vsetrangepoints": 5, "expectedvariablenames": ["x"], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "2+8{v}/x^3", "marks": 1, "checkvariablenames": true, "checkingtype": "absdiff", "vsetrange": [0, 1]}, {"vsetrangepoints": 5, "expectedvariablenames": ["Y", "N", "U"], "checkingaccuracy": 0.001, "type": "jme", "showpreview": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "Y", "marks": 1, "checkvariablenames": true, "checkingtype": "absdiff", "vsetrange": [0, 1]}, {"precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "precision": "2", "maxValue": "{a}^2+4({v}/{a})", "minValue": "{a}^2+4({v}/{a})", "variableReplacementStrategy": "originalfirst", "strictPrecision": false, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": "50", "scripts": {}, "marks": "1", "type": "numberentry", "showPrecisionHint": false}], "steps": [{"prompt": "

Hint: find $x$ at stationary point

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "marks": 0, "scripts": {}, "showCorrectAnswer": true, "type": "gapfill"}], "statement": "

An open metal tank of square base has a volume of $\\var{v}\\text{ m}^3$

\n

Given that the square base has sides of length $x$ metres, find expressions, in terms of $x$, for the following.

", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"a": {"definition": "siground((2v)^(1/3),3)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "v": {"definition": "random(100..200)", "templateType": "anything", "group": "Ungrouped variables", "name": "v", "description": ""}}, "metadata": {"notes": "", "description": "

Another practical application of differentiation

", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Katie Lester", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/586/"}]}]}], "contributors": [{"name": "Katie Lester", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/586/"}]}