Mathematics, 02.11.2019 05:31 dixonmckenzie1429
Asprinkler distributes water in a circular pattern, supplying water to a depth of e^-r feet per hour at a distance of r feet from the sprinkler.
a) what is the total amount of water supplied per hour inside of a circle of radius 11?
b) what is the total amount of water that goes throughout the sprinkler per hour?
Answers: 1
![W(R=11)=\int\limits^R_{-R}\int\limits^R_{-R} {f(x,y)} \, dxdy = \int\limits^{2\pi} _{0}\int\limits^R_{0} {f(x,y)} \, r drd\alpha = \int\limits^{2\pi} _{0}\int\limits^R_{0} e^{-r} \, r drd\alpha = 2\pi [2-(R+1)e^{-R} ] = 2\pi [2-(11+1)e^{-11} ] = 12,5651](/tpl/images/0356/5411/ec308.png)
![W(R=\infty)= \lim_{R \to \infty} 2\pi [2-(R+1)e^{-R} ] =2\pi [2- \lim_{R \to \infty}(R+1)e^{-R} ] = 2\pi [2- \lim_{R \to \infty}(R+1)/e^{R} ] = 2\pi [2- \lim_{R \to \infty} 1/e^{R} ] = 2\pi [2- 0] = 4\pi = 12,5663](/tpl/images/0356/5411/3b018.png)
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Asprinkler distributes water in a circular pattern, supplying water to a depth of e^-r feet per hour...
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