Answer to Question #156045 in Trigonometry for Anne

Question #156045

We want to build a greenhouse that has a half cylinder roof of radius r and height r mounted horizontally on top of four rectangular walls of height h as shown in the figure. We have 200π m2 of plastic sheet to be used in the construction of this structure. Find the value of r for the greenhouse with the largest possible volume we can build.



1
Expert's answer
2021-01-18T16:48:42-0500

"V=2r^2+\\frac{\\pi r^2h}{2};" "S_{surface}=2\\pi r^2+2r^2+6rh;" We applied differentiation like this:


1) "\\frac{dV}{dh}+\\frac{dV}{dr}=0:"


"\\frac{\\pi r^2}{2}\\frac{dV}{dh}+4r+ \\pi rh=0;"


"\\frac{dV}{dh}=-\\frac{2(4+\\pi h)}{\\pi r};"


2) "\\frac{dS}{dh}+\\frac{dS}{dr}=0:"

"6r\\frac{dS}{dh}+4\\pi r+4r+6h=0;"


"6r(-\\frac{2(4+\\pi h)}{\\pi r})+4\\pi r+4r+6h=0;"


"-12(4+\\pi h)+4\\pi^2 r+4\\pi r+6\\pi h=0;"


"-48-12\\pi h+4\\pi^2 r+4\\pi r+6\\pi h=0;"


"-48-6\\pi h+4\\pi^2 r+4\\pi r=0;"

"h=\\frac{2\\pi^2 r+2\\pi r-24}{3\\pi};"


There is given "S=\\pi rh=200" :


"\\pi r\\times\\frac{2\\pi^2 r+2\\pi r-24}{3\\pi}=200;"


"r\\times(\\pi^2 r+\\pi r-12)=300;"


"r\\times(r\\times(1+\\frac{1}{\\pi})-\\frac{12}{\\pi^2})=300;"


"r_1=\\frac{12+\\sqrt{144+1200\\pi ^4+1200\\pi ^3}}{2\\pi ^2+2\\pi }" "r_2=\\frac{12-\\sqrt{144+1200\\pi ^4+1200\\pi ^3}}{2\\pi ^2+2\\pi }"


Here is "r_2" is negative. So r would get the value of "r_1" . And can find V:


"V=2r^2+\\frac{\\pi r^2h}{2};"


"V=2r^2+\\frac{\\pi r^2 \\times\\frac{2\\pi^2 r+2\\pi r-24}{3\\pi}}{2};"


"V=2r^2+\\frac{ r^2\\times(2\\pi^2 r+2\\pi r-24)}{6};"


"V=r^3(\\frac{ \\pi^2 +\\pi }{3})-2r^2=0;"

"V=r^2\\times(r\\times(\\frac{ \\pi^2 +\\pi }{3})-2)=0;" where, "r=\\frac{12+\\sqrt{144+1200\\pi ^4+1200\\pi ^3}}{2\\pi ^2+2\\pi };"


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment

LATEST TUTORIALS
New on Blog
APPROVED BY CLIENTS