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{"name": "mass of tetrahedron", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "mass of tetrahedron", "tags": [], "metadata": {"description": "<p>mass of tetrahedron via integration</p>", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "<p>Cosider the tetrahedron bounded by the coordinate planes $x=0$, $y=0$ and $z=0$ and the plane $\\frac{x}{\\var{a}} + \\frac{y}{\\var{b}} + \\frac{z}{\\var{c}} =1$. Assume that it has a density function $\\rho(x) = \\var{R0}(\\var{a}+x)$.&nbsp;</p>", "advice": "<p><strong>a)</strong></p>\n<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>We can write the boundaries of the tetrahedron as&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) \\quad\\mbox{and }\\quad 0\\leq z \\leq \\var{c}\\left( 1 - \\frac{x}{\\var{a}} - \\frac{y}{\\var{b}}\\right).\\]</p>\n<p></p>\n<p>Thus, with mass element $dM = \\rho(x)dV = \\rho(x)dx\\,dy\\,dz$, the total mass is&nbsp;</p>\n<p>\\[M= \\int_T\\,dM = \\int_0^\\var{a}\\int_0^{\\frac{\\var{b}}{\\var{a}}(\\var{a} -x)}\\int_0^{\\var{c}\\left( 1 - \\frac{x}{\\var{a}} - \\frac{y}{\\var{b}}\\right)}\\rho(x)\\,dz\\,dy\\,dx =\\\\[3mm]<br/>&nbsp;\\int_0^\\var{a}\\int_0^{\\frac{\\var{b}}{\\var{a}}(\\var{a} -x)}\\int_0^{\\var{c}\\left( 1 - \\frac{x}{\\var{a}} - \\frac{y}{\\var{b}}\\right)} \\var{R0}(\\var{a}+x) \\,dz\\,dy\\,dx =\\\\[3mm]<br/>\\int_0^\\var{a} \\var{R0}(\\var{a}+x) \\int_0^{\\frac{\\var{b}}{\\var{a}}(\\var{a} -x)}\\int_0^{\\var{c}\\left( 1 - \\frac{x}{\\var{a}} - \\frac{y}{\\var{b}}\\right)}&nbsp; \\,dz\\,dy\\,dx =\\\\[3mm]<br/>\\int_0^\\var{a} \\var{R0}(\\var{a}+x) \\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{R0}(\\var{a}+x)\\left[ \\frac{\\simplify{{b*c}}}{\\simplify{{2*a^2}}}(\\var{a}-x)^2\\right]\\, dx = \\frac{5\\cdot\\var{a}^2\\cdot\\var{b}\\cdot\\var{c}\\cdot\\var{R0}}{24}=\\frac{\\simplify{{5*{a^2}*{b}*{c}*{R0}}}}{24}.\\]</p>\n<p></p>\n<p><strong>b)</strong></p>\n<p>We start with the $x$ coordinate.</p>\n<p>\\[\\bar x= \\frac{1}{M}\\int_T x\\,dM = \\int_0^\\var{a}\\int_0^{\\frac{\\var{b}}{\\var{a}}(\\var{a} -x)}\\int_0^{\\var{c}\\left( 1 - \\frac{x}{\\var{a}} - \\frac{y}{\\var{b}}\\right)}x\\rho(x)\\,dz\\,dy\\,dx =\\\\[3mm]<br/>\\frac{1}{M} \\int_0^\\var{a}\\int_0^{\\frac{\\var{b}}{\\var{a}}(\\var{a} -x)}\\int_0^{\\var{c}\\left( 1 - \\frac{x}{\\var{a}} - \\frac{y}{\\var{b}}\\right)} x\\var{R0}(\\var{a}+x) \\,dz\\,dy\\,dx =\\\\[3mm]<br/>\\frac{1}{M}\\int_0^\\var{a} x\\var{R0}(\\var{a}+x) \\int_0^{\\frac{\\var{b}}{\\var{a}}(\\var{a} -x)}\\int_0^{\\var{c}\\left( 1 - \\frac{x}{\\var{a}} - \\frac{y}{\\var{b}}\\right)}&nbsp; \\,dz\\,dy\\,dx =\\\\[3mm]<br/>\\frac{1}{M}\\int_0^\\var{a} x\\var{R0}(\\var{a}+x) \\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/>\\frac{1}{M}\\int_0^\\var{a} x\\var{R0}(\\var{a}+x)\\left[ \\frac{\\simplify{{b*c}}}{\\simplify{{2*a^2}}}(\\var{a}-x)^2\\right]\\, dx =\\\\[3mm]<br/>\\frac{\\simplify{{R0*b*c}}}{\\simplify{{2*a^2}}M}\\int_0^\\var{a} x&nbsp; (\\var{a}+x)\\left[ (\\var{a}-x)^2\\right]\\, dx =<br/>\\frac{\\simplify{{R0*b*c}}}{\\simplify{{2*a^2}}M}\\int_0^\\var{a}x(\\var{a^3} -\\var{a^2}x-\\var{a} x^2 + x^3) \\, dx =\\\\[3mm]<br/>\\frac{\\simplify{{R0*b*c}}}{\\simplify{{2*a^2}}M}\\int_0^\\var{a}\\var{a^3}x -\\var{a^2}x^2-\\var{a} x^3 + x^4 \\, dx = \\\\[3mm]<br/>\\frac{\\simplify{{R0*b*c}}}{\\simplify{{2*a^2}}M}\\left[\\frac{\\var{a^3}x^2}{2} -\\frac{\\var{a^2}x^3}{3}-\\frac{\\var{a} x^4}{4} + \\frac{x^5}{5} \\right]_0^{\\var{a}}=\\\\[3mm]<br/>\\simplify[fractionNumbers]{({a}^5/2 - ({a}^5)/3 - {a}^5/4 + {a}^5/5) *{R0*b*c}/{2*a^2*Mass}} = <br/>\\simplify[fractionNumbers,otherNumbers]{({a}^5/2 - ({a}^5)/3 - {a}^5/4 + {a}^5/5) *{R0*b*c}/{2*a^2*Mass}}.\\]</p>\n<p>Simlarly, for $y$ coordinate:</p>\n<p>\\[\\bar y = \\frac{1}{M}\\int_T y\\,dM = \\int_0^\\var{a}\\int_0^{\\frac{\\var{b}}{\\var{a}}(\\var{a} -x)}\\int_0^{\\var{c}\\left( 1 - \\frac{x}{\\var{a}} - \\frac{y}{\\var{b}}\\right)}y\\rho(x)\\,dz\\,dy\\,dx =\\\\[3mm]<br/>\\frac{1}{M} \\int_0^\\var{a}\\int_0^{\\frac{\\var{b}}{\\var{a}}(\\var{a} -x)}\\int_0^{\\var{c}\\left( 1 - \\frac{x}{\\var{a}} - \\frac{y}{\\var{b}}\\right)} y\\var{R0}(\\var{a}+x) \\,dz\\,dy\\,dx =\\\\[3mm]<br/>\\frac{1}{M}\\int_0^\\var{a} \\var{R0}(\\var{a}+x) \\int_0^{\\frac{\\var{b}}{\\var{a}}(\\var{a} -x)} y \\int_0^{\\var{c}\\left( 1 - \\frac{x}{\\var{a}} - \\frac{y}{\\var{b}}\\right)}&nbsp; \\,dz\\,dy\\,dx =\\\\[3mm]<br/>\\frac{1}{M}\\int_0^\\var{a} \\var{R0}(\\var{a}+x) \\int_0^{\\frac{\\var{b}}{\\var{a}}(\\var{a} -x)} y\\var{c}\\left( 1 - \\frac{x}{\\var{a}} - \\frac{y}{\\var{b}}\\right) \\,dy\\,dx =\\\\[3mm]<br/>\\frac{1}{M}\\int_0^\\var{a} \\var{R0}(\\var{a}+x) \\int_0^{\\frac{\\var{b}}{\\var{a}}(\\var{a} -x)} \\var{c}\\left( y - \\frac{xy}{\\var{a}} - \\frac{y^2}{\\var{b}}\\right) \\,dy\\,dx =\\\\[3mm]<br/>\\frac{1}{M}\\int_0^\\var{a} \\var{R0}(\\var{a}+x)\\frac{\\simplify{{c*b^2}}}{\\simplify{{6*a^3}}}\\left[\\var{a^3}-3\\cdot\\simplify[unitFactor]{{a^2}x} +3\\cdot\\simplify[unitFactor]{{a}x^2}-x^3\\right]\\, dx =\\\\[3mm]<br/>\\frac{\\simplify{{R0*b^2*c}}}{\\simplify{{6*a^3}}M}\\int_0^\\var{a}&nbsp; &nbsp;(\\var{a}+x)( \\var{a}-x)^3, dx =<br/>\\frac{\\simplify{{R0*b^2*c}}}{\\simplify{{6*a^3}}M}\\int_0^\\var{a}\\var{a^4} -2\\cdot\\simplify[unitFactor]{{a^3}x} +2\\cdot\\simplify[unitFactor]{{a}x^3}-x^4 \\, dx =\\\\[3mm]<br/>\\frac{\\simplify{{R0*b^2*c}}}{\\simplify{{6*a^3}}M}\\left[\\simplify[unitFactor]{{a^4}x} - \\simplify[unitFactor]{{a^3}x^2} +\\frac{\\simplify[unitFactor]{{a}x^4}}{2}-\\frac{x^5}{5} \\right]_0^\\var{a} = \\\\[3mm]<br/><br/>\\simplify[fractionNumbers]{({a}^5 - ({a}^5) + {a}^5 /2- {a}^5/5) *{R0*b^2*c}/{6*a^3*Mass}} = <br/>\\simplify[fractionNumbers,otherNumbers]{({a}^5 - ({a}^5) + {a}^5 /2- {a}^5/5) *{R0*b^2*c}/{6*a^3*Mass}}.\\]</p>\n<p>Finaly, for $z$ coordinate:</p>\n<p>\\[\\bar z= \\frac{1}{M}\\int_T z\\,dM = \\int_0^\\var{a}\\int_0^{\\frac{\\var{c}}{\\var{a}}(\\var{a} -x)}\\int_0^{\\var{b}\\left( 1 - \\frac{x}{\\var{a}} - \\frac{z}{\\var{c}}\\right)}z\\rho(x)\\,dy\\,dz\\,dx =\\\\[3mm]<br/>\\frac{1}{M}\\int_0^\\var{a}\\int_0^{\\frac{\\var{c}}{\\var{a}}(\\var{a} -x)}\\int_0^{\\var{b}\\left( 1 - \\frac{x}{\\var{a}} - \\frac{z}{\\var{c}}\\right)} z\\var{R0}(\\var{a}+x) \\,dy\\,dz\\,dx =\\\\[3mm]<br/>\\frac{1}{M} \\int_0^\\var{a}\\var{R0}(\\var{a}+x)\\int_0^{\\frac{\\var{c}}{\\var{a}}(\\var{a} -x)}z\\int_0^{\\var{b}\\left( 1 - \\frac{x}{\\var{a}} - \\frac{z}{\\var{c}}\\right)}&nbsp; &nbsp;\\,dy\\,dz\\,dx =\\\\[3mm]<br/>\\frac{1}{M}\\int_0^\\var{a} \\var{R0}(\\var{a}+x)\\int_0^{\\frac{\\var{c}}{\\var{a}}(\\var{a} -x)} z\\var{b}\\left( 1 - \\frac{x}{\\var{a}} - \\frac{z}{\\var{c}}\\right)\\,dz\\,dx =\\\\[3mm]<br/>\\frac{1}{M}\\int_0^\\var{a} \\var{R0}(\\var{a}+x)\\int_0^{\\frac{\\var{c}}{\\var{a}}(\\var{a} -x)} \\var{b}\\left( z - \\frac{xz}{\\var{a}} - \\frac{z^2}{\\var{c}}\\right)\\,dz\\,dx =\\\\[3mm]<br/>\\frac{\\simplify{{R0*a}}}{M}\\int_0^\\var{a} (\\var{a}+x) \\left[ \\frac{z^2}{2} - \\frac{xz^2}{\\var{2*a}} - \\frac{z^3}{\\var{3*c}}\\right]_0^{\\frac{\\var{c}}{\\var{a}}(\\var{a} -x)}\\,dx =\\\\[3mm]<br/>\\frac{\\simplify{{R0*c^2}}}{\\simplify{{6*a^2}}M}\\int_0^\\var{a} (\\var{a}+x)(a-x)^3\\,dx = <br/>\\frac{\\simplify{{R0*c^2}}}{\\simplify{{6*a^2}}M}\\int_0^\\var{a} \\var{a^4}-\\var{2*a^3}x + \\var{2*a}x^3 - x^4\\,dx\\\\[3mm]<br/>\\frac{\\simplify{{R0*c^2}}}{\\simplify{{6*a^2}}M}\\left[\\var{a^4}x -\\frac{\\var{2*a^3}x^2}{2} + \\frac{\\var{a}x^4}{2} - \\frac{x^5}{5}\\right]_0^{\\var{a}}=\\\\[3mm]<br/><br/><br/><br/><br/>\\simplify[fractionNumbers]{({a^5} -{2*a^5}/2 + {a^5}/2 - {a^5}/5) *{R0*c^2}/{6*a^2*Mass}} = <br/>\\simplify[fractionNumbers,otherNumbers]{({a^5} -{2*a^5}/2 + {a^5}/2 - 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