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Thank you
a)
we know,
volume of cone = π r2 h /3
where r = radius of cone and h= height of cone
given,
height = 3 inches , radius = 1 inch
now,
at an instant , height = y
to find radius at that instant , use property of similar triangles .
which gives,
3/y = 1/r
r= y/3
so we, got height and radius at that instant .
therefore,
volume at that instant , V = π r2 h /3 = π *(y/3)2 *y / 3 = π y3 /27.
b)
given,
height of sand in bottom cone at that instant = h
therefore, height of empty space in cone = 3-h
to find radius of empty space at that instant , use property of similar triangles .
which gives,
(3-h)/3 = r/1
r= (3-h)/3
therefore, volume of empty space in cone B = V= π r2 h /3 = π *[(3-h)/3]2 (3-h) /3 =π *(3-h)3/27 .
now , volume of sand in cone B = total volume of cone B - volume of empty space
= π r2 h /3 - π *(3-h)3/27
= π 12*3/3 -π *(3-h)3/27 .
= π - π *(3-h)3/27 .
c)
total volume of sand = 3π/4.
total volume of sand = volume of sand in A + volume of sand in B
3 π/4 = π y3 /27 + π - π *(3-h)3/27
at an instant , h= 1 (given), now find y at that instant
therefore,
3 π/4 = π y3 /27 + π - π *(3-1)3/27
3 π/4 = π y3 /27 + π - π *(2)3/27
3 π/4 = π y3 /27 + π - π *8/27
3 π/4 = π y3 /27 + π 19/27
27* 3 π/4 = π y3 + 19 π
cancelling π
81/4 = y3 +19
y3 = 81/4 -19
y3 = 5/4.
y = 1.077.
now,
volume of uppercone A = V = π y3 /27.
derievating
dV = 3 π y2 dy /27
= π y2 dy /9
dy = 2/100 = 0.02 , y( we got it) = 1.077
therefore,
dV = π (1.077)2 *0.02 /9
= 0.002578 π
it is true that rate of chane of volume for both the cylinders is equal
therefore,
dV for cone B = 0.002578 π
now,
we have ,
volume of cone B = V = π - π *(3-h)3/27
differentiating,
dV = - π 3*(-1)*(3-h)2 dh/27
at this instant, h=1
0.002578 π = π *(3-1)2 dh /9
0.002578 π = π *22 dh /9
0.002578 π = π *4dh /9
cancelling π
0.002578 = 4dh /9
dh = 9*0.002578/4
= 0.0058 inches per second.
= 58/10000
rate of increase in height h in cone B = 0.0058 inches per second.
note: i am sorry i dont have a good camera phone . so i typed it here. even the code cogs equation editor is not working here. otherwise , it would have been better.
i hope you will understand it.
if you have any doubts just comment below . i will get back to you as soon as possible.
thank you :)
Please solve on a piece of paper then upload it ( show all steps) Thank you...
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