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marigirl
 one year ago
Rate of change Differentiation
An inverted cone with a base area of 2500pi mm^2 and height of 300m is slowly being filled with water. Water is being poured in at a rate of 350 mm^3/s. At what rate is the height of the water changing when the water is 100mm from the top of the inverted cone?
marigirl
 one year ago
Rate of change Differentiation An inverted cone with a base area of 2500pi mm^2 and height of 300m is slowly being filled with water. Water is being poured in at a rate of 350 mm^3/s. At what rate is the height of the water changing when the water is 100mm from the top of the inverted cone?

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marigirl
 one year ago
Best ResponseYou've already chosen the best response.1I think: We are to find out: \[\frac{ dh }{ dt } \] We know \[\frac{ dV }{ dt }=350\]

marigirl
 one year ago
Best ResponseYou've already chosen the best response.1dw:1443476761216:dw

marigirl
 one year ago
Best ResponseYou've already chosen the best response.1Volume of cone: \[V=\frac{ 1 }{ 3 }\pi r^2h\] Volume of this cone:\[V=\frac{ 2500 }{ 3 }h\]

phi
 one year ago
Best ResponseYou've already chosen the best response.1use "similar triangles" to write r in terms of h

phi
 one year ago
Best ResponseYou've already chosen the best response.1from the area of the base you find r=50 and you have this dw:1443477912255:dw

phi
 one year ago
Best ResponseYou've already chosen the best response.1\[ V=\frac{ 1 }{ 3 }\pi r^2h \\ V=\frac{ 1 }{ 3 }\pi \frac{h^2}{36}h \\ V= \frac{\pi}{108} h^3 \] now use implicit differentiation

marigirl
 one year ago
Best ResponseYou've already chosen the best response.1\[\frac{ dv }{ dh }=36 \pi h^2\] \[\frac{ dh }{ dt }=\frac{ dv }{ dt }\times \frac{ dh }{ dv }\] am i on the right track

marigirl
 one year ago
Best ResponseYou've already chosen the best response.1then \[\frac{ dh }{ dt }=350 \times \times \frac{ 1 }{ 36 \pi h^2 }\] and then sub in h=100

phi
 one year ago
Best ResponseYou've already chosen the best response.1I would take the derivative of each variable with respect to t on the left side you get dv/dt on the right side you get \[\frac{\pi}{108} \frac{d}{dt} h^3 \] which is \[ \frac{\pi}{108} 3 h^2\frac{d}{dt}h\] so the equation is \[ \frac{dV}{dt}= \frac{\pi}{36} h^2 \frac{dh}{dt} \]

phi
 one year ago
Best ResponseYou've already chosen the best response.1when the water is 100mm from the top of the inverted cone in the formula h is the height from 0 up to 300 100 mm from the top corresponds to h= 200 mm

phi
 one year ago
Best ResponseYou've already chosen the best response.1Your approach works except that this ***\( \frac{ dv }{ dh }=36 \pi h^2 \)*** should read \[ \frac{dV}{dh}= \frac{\pi}{36} h^2\]

marigirl
 one year ago
Best ResponseYou've already chosen the best response.1then sub in h=200....
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