a sprinkler 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 16? ft3 per hour b. what is the total amount of water that goes throught the sprinkler per hour? ft3 per hour calculate the total amount of water supplied per hour inside a circle of radius r, then let r→[infinity]. note: you can earn partial credit on this problem.

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30-08-2022
Mathematics

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a sprinkler 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 16? ft3 per hour b. what is the total amount of water that goes throught the sprinkler per hour? ft3 per hour calculate the total amount of water supplied per hour inside a circle of radius r, then let r→[infinity]. note: you can earn partial credit on this problem.

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tripatsingh039 tripatsingh039 · Answer 1 · 2022-09-04T05:53:52-04:00

A)The sum of all the water distributed from 0 to r is 2[tex]\pi[/tex][tex](2-\frac{17}{e^{16} })[/tex]

B)The amount of the water at all the distribution from 0 to ∞ is 4[tex]\pi[/tex]

Given that f(r)=[tex]e^{-r}[/tex] ; asked to calculate the sum of all the water distributed from 0 to r and sum of all the water distributed from 0 to ∞.

The sum of all the water distributed from 0 to r= 2[tex]\pi[/tex][ 2-(R+1)[tex]e^{-R}[/tex]]

The sum of all the water distributed from 0 to r= 2[tex]\pi[/tex][ 2-(16+1)[tex]e^{-16}[/tex]]

The sum of all the water distributed from 0 to r= 2[tex]\pi[/tex][tex](2-\frac{17}{e^{16} })[/tex]

The amount of the water at all the distribution from 0 to ∞=2[tex]\pi[/tex][2-[tex]\lim_{R \to \infty} \frac{R+1}{e^{R} }[/tex]]

The amount of the water at all the distribution from 0 to ∞=2[tex]\pi[/tex][2-[tex]\lim_{R \to \infty} \frac{1}{e^{R} }[/tex]]

The amount of the water at all the distribution from 0 to ∞=4[tex]\pi[/tex].

Therefore,A)The sum of all the water distributed from 0 to r is 2[tex]\pi[/tex][tex](2-\frac{17}{e^{16} })[/tex]

B)The amount of the water at all the distribution from 0 to ∞ is 4[tex]\pi[/tex]

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