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{"name": "Volume of a tetrahedron", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Volume of a tetrahedron", "tags": [], "metadata": {"description": "<p>Volume of a tetrahedron using integrals</p>", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "<p>Find the volume of the tetrahedron bounded by the coordinate planes $x=0$, $y=0$ and $z=0$ and the plane $\\frac{x}{\\var{a}} + \\frac{y}{\\var{b}} + \\frac{z}{\\var{c}} =1$.</p>", "advice": "<p>The coordinate surfaces $x=0$, $y=0$ and $z=0$ are given. The plane $\\frac{x}{\\var{a}} + \\frac{y}{\\var{b}} + \\frac{z}{\\var{c}} = 1$ can be rewritten as a functions as follows&nbsp;</p>\n<p>\\[z = f(x,y) = \\var{c}\\left( 1 - \\frac{x}{\\var{a}} - \\frac{y}{\\var{b}}\\right).\\]</p>\n<p>The surface of $f(x,y)$ intersects the axes at $x=\\var{a}, y=\\var{b}$ and $z=\\var{c}$.</p>\n<p>Let us integrate $y$-firts. Then the triangular region the tetrahedron sits is bounded by the lines $x=0, y=0$ and $\\frac{x}{\\var{a}} + \\frac{y}{\\var{b}} = 1$. Then we have the following bounds for integration&nbsp;</p>\n<p>\\[0&lt;x&lt;\\var{a} \\quad \\mbox{and} \\quad 0 &lt; y&lt;&nbsp; \\frac{\\var{b}}{\\var{a}}(\\var{a} -x).\\]</p>\n<p>Hence, we can compute the volume of the tetrahedron as&nbsp;</p>\n<p>\\[V = \\int_0^\\var{a}\\int_0^{\\frac{\\var{b}}{\\var{a}}(\\var{a} -x)}f(x,y) \\,dy\\, dx = \\int_0^\\var{a}\\int_0^{\\frac{\\var{b}}{\\var{a}}(\\var{a} -x)} \\var{c}\\left( 1 - \\frac{x}{\\var{a}} - \\frac{y}{\\var{b}}\\right) \\,dy\\, dx =\\\\[3mm]<br/>= \\int_0^\\var{a} \\var{c}\\left[ y - \\frac{xy}{\\var{a}} - \\frac{y^2}{\\simplify{2*{b}}}\\right]_0^{\\frac{\\var{b}}{\\var{a}}(\\var{a} -x)}\\, dx = \\int_0^\\var{a} \\frac{\\simplify{{b*c}}}{\\simplify{{2*a^2}}}(\\var{a}-x)^2 \\, dx = \\frac{\\simplify{{a*b*c}}}{6}\\]</p>\n<pre></pre>\n<pre></pre>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1 .. 6#1)", "description": "", "templateType": "randrange", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(1 .. 6#1)", "description": "", "templateType": "randrange", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(1 .. 6#1)", "description": "", "templateType": "randrange", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "c"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>(enter your answer as an interger or a fraction)</p>", "answer": "{a}*{b}*{c}/6", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Ugur Efem", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18261/"}]}]}], "contributors": [{"name": "Ugur Efem", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18261/"}]}