Question

I hope there is someone there who can solve this task:
A tank is formed like a cone with the tip pointing downwards. It has the radius 10 dm and height is 20 dm. we are leading water into the tank with an inflowspeed at 1 liter per minute. Calculate the growth rate of the areal of the watersurface when the waterdepth is 8 dm.

Please show how to do this.

A tank is formed like a cone with the tip pointing downwards. It has the radius 10 dm and height is 20 dm. we are leading water into the tank with an inflowspeed at 1 liter per minute. Calculate the growth rate of the areal of the watersurface when the waterdepth is 8 dm.

Answer 1

The detailed solution is given below:

Given 10-0 cone = h = 20 dm. higher of the - 200 cm. and radies of the base= 10 dn. 100 din, Diameter of the base height This relation holds im each depth of the Now, the imflow 1 eor , min. -100 0 cm 3. mom I we have to find the hale Howta سی of areal the water surface when depth is 8 dn = 80 cm. 14 water surface by we define volume = va area of the 13 have to find dA at 500 A we h =BO wenn know Cone Volume 13 area of the er face x heighn oty А her tople take Anford bready forex A Simee find while - have to Ао we DA at will losslag i in )( لیا terms of A im RHS of (2), Now, have ule that established on this case already the height will always be equal to diameter als o area TM² where = nading, Then A - T42 ТҮС h = 2n T/A 4. or h = NA

hane دوا Putting this value h of in (i) 113 X A DT 3/2 (*) ory 2/31 with respect to Now, taking derivative of both sides have 12 319 A dA atti .at A = A (t) j О au 2 at dt wib volume = infloor hate Here du means rate of change of at 3 min, 1000 can dA rowth rate role serface (gre of change of at have to dA h = 80 cm. So, ha 80 cm TN (h 80 (40)2 these Peutting values im (ui) havie we - 1 looo 7.69.02 (4092 da da od at oty dA 1000 at 40 Hence have evaluate d we the growth rate of areal of 25 cm²1 the surface which 1 min.