Answer :
Answer:
376966.991 Joules
Explanation:
Given that :
the height = 12 m
Let assume the tank have a thickness = dh
The radius of the tank by using the concept of similar triangle is :
[tex]\dfrac{1}{r} = \dfrac{12}{h}[/tex]
[tex]r = \dfrac{h}{12}[/tex]
The area of the tank = [tex]\mathbf{\pi r^2}[/tex]
The area of the tank = [tex]\mathbf{\pi( \dfrac{h}{12})^2}[/tex]
The area of the tank = [tex]\mathbf{ \dfrac{\pi}{144}h^2}[/tex]
The volume of the tank is = area × thickness
= [tex]\mathbf{ \dfrac{\pi}{144}h^2 \ dh}[/tex]
Weight of the element = [tex]\rho_ g * volume[/tex]
where;
[tex]\rho_g[/tex] = density of water ; which is given as 10000 N/m³
So;
Weight of the element = [tex]\mathbf{ 10000 *\dfrac{\pi}{144}h^2 \ dh}[/tex]
Weight of the element = [tex]\mathbf{69.44 \ \pi \ h^2 \ dh}[/tex]
However; the work required to pump this water = weight × height rise
where the height rise = 12 - h
the work required to pump this water = [tex]\mathbf{69.44 \ \pi \ h^2 \ dh}[/tex](12 - h)
the work required to pump this water = [tex]\mathbf{69.44 \pi (12h^2-h^3)dh}[/tex]
We can determine the total workdone by integrating the work required to pump this water
SO;
Workdone = [tex]\mathbf{\int\limits^{12}_0 {69.44 \pi(12h^2-h^3)} dh}[/tex]
= [tex]\mathbf{ 69.44 \pi \int\limits^{12}_0 {(12h^2-h^3)} dh}[/tex]
= [tex]\mathbf{ 69.44 \pi[ \frac{12h^3}{3}- \frac{h^4}{4}]^{12}}_0} }[/tex]
= [tex]\mathbf{69.44 \pi [ \frac{12^4}{3}-\frac{12^4}{4}]}[/tex]
= [tex]\mathbf{69.44 \pi*12^4 [ \frac{4-3}{12}]}[/tex]
= [tex]\mathbf{69.44 \pi*12^4 *\frac{1}{12}}[/tex]
= 376966.991 Joules
The required work done to pump all the water from the tank would be as follows:
[tex]376966.991[/tex] Joules
Find the Work Done?
Given that,
Height [tex](h)[/tex] [tex]= 12 meter[/tex]
Assuming the thickness [tex]= dh[/tex]
Radius through 'similar triangle concept' would be:
[tex]r = h/12[/tex]
Now,
The tank's area would be [tex]=[/tex] π[tex]r^2[/tex]
[tex]=[/tex] π[tex](h/12)^2[/tex]
[tex]=[/tex] (π[tex]/144)[/tex][tex]h^2[/tex]
The tank's volume [tex]=[/tex] Area × Thickness
[tex]=[/tex] (π[tex]/144)[/tex][tex]h^2[/tex] [tex]dh[/tex]
Now,
The element's weight can be determined through [tex]p_{g}[/tex] × [tex]volume[/tex]
where [tex]p_{g}[/tex] [tex]= 1000 N[/tex]
So,
Element's weight [tex]= 1000[/tex] × (π[tex]/144)[/tex][tex]h^2[/tex] [tex]dh[/tex]
[tex]= 69.44[/tex] π[tex]h^2 dh[/tex]
So,
The work required to pump all the water can be determined through
[tex]weight[/tex] × [tex]height rise[/tex]
where
[tex]height rise = 12 - h[/tex]
Therefore,
The work required to pump all the water
[tex]= 69.44[/tex] π[tex]h^2 dh[/tex] [tex](12 - h)[/tex]
[tex]= 69.44[/tex] π[tex](12h^2 - h^3)dh[/tex]
Now, after integration, the Total work done
[tex]=[/tex][tex]\int\limits^ \,12[/tex][tex]_{0}[/tex] [tex]69.44[/tex]π[tex](12h^2 - h^3)dh[/tex]
[tex]= 69.44[/tex] π × [tex]12^4[/tex] × [tex]1/12[/tex]
[tex]=[/tex] [tex]376966.991[/tex] Joules
Thus, [tex]376966.991[/tex] Joules is the correct answer.
Learn more about "Work Done" here:
brainly.com/question/3902440