Anthonyk2013
Junior Member
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- Sep 15, 2013
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Struggling with Q 5
5. A closed cylindrical container has a surface area of 400 cm2. Determine the dimensions of maximum volume.
My work:
\(\displaystyle V\, =\, \pi r^2 h\)
\(\displaystyle \mbox{Surface area: }\, 2 \pi r h\, 2 \pi r^2\, =\, 400\)
\(\displaystyle V\, =\, \pi r^2\, \left(\dfrac{400\, -\, 2 \pi r^2}{2 \pi r}\right)\)
\(\displaystyle V\, =\, r\, \left(\dfrac{400\, -\, 2 \pi r^2}{2}\right)\)
\(\displaystyle V\, =\, r\, (200\, -\, \pi r^2)\)
\(\displaystyle V\, =\, 200r\, -\, \pi r^2\)
\(\displaystyle \dfrac{dV}{dr}\, =\, 200\, -\, 2 \pi r\)
\(\displaystyle 200\, -\, 2 \pi r\, =\, 0\)
\(\displaystyle 200\, =\, 2 \pi r\)
\(\displaystyle \dfrac{200}{2 \pi}\, =\, r\)
\(\displaystyle r\, =\, 31.83\)
\(\displaystyle \dfrac{d^2V}{dr^2}\, =\, 0\, -\, 2 \pi\)
5. A closed cylindrical container has a surface area of 400 cm2. Determine the dimensions of maximum volume.
My work:
\(\displaystyle V\, =\, \pi r^2 h\)
\(\displaystyle \mbox{Surface area: }\, 2 \pi r h\, 2 \pi r^2\, =\, 400\)
\(\displaystyle V\, =\, \pi r^2\, \left(\dfrac{400\, -\, 2 \pi r^2}{2 \pi r}\right)\)
\(\displaystyle V\, =\, r\, \left(\dfrac{400\, -\, 2 \pi r^2}{2}\right)\)
\(\displaystyle V\, =\, r\, (200\, -\, \pi r^2)\)
\(\displaystyle V\, =\, 200r\, -\, \pi r^2\)
\(\displaystyle \dfrac{dV}{dr}\, =\, 200\, -\, 2 \pi r\)
\(\displaystyle 200\, -\, 2 \pi r\, =\, 0\)
\(\displaystyle 200\, =\, 2 \pi r\)
\(\displaystyle \dfrac{200}{2 \pi}\, =\, r\)
\(\displaystyle r\, =\, 31.83\)
\(\displaystyle \dfrac{d^2V}{dr^2}\, =\, 0\, -\, 2 \pi\)
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