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document.write( "A drawing of a cross section of the cone, might be helpful for tutor Ikleyn's\r\n" );
document.write( "solution:\r\n" );
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document.write( "The part of the conical tank, occupied by water, is the cone with the ratio radius to the height\r\n" );
document.write( "of 
= 
= 0.6.\r\n" );
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document.write( "This is because of corresponding parts of similar right triangles.\r\n" );
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document.write( "Hence, the volume of the water in the tank at every time moment t is\r\n" );
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document.write( " V(t) = 
= 
= 
. (1)\r\n" );
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document.write( "The mistake some of my students were prone to make at this point was their \r\n" );
document.write( "failure to realize that h is a VARIABLE, and not the CONSTANT 3. Some of\r\n" );
document.write( "my students would try to substitute 3 for h at this point, and didn't\r\n" );
document.write( "know what to do next. But we must first allow h to vary, and then only\r\n" );
document.write( "\"freeze\" the variable at h=3 AFTERWARDS. We must NOT substitute 3 for h\r\n" );
document.write( "until after we have employed differentiation. \r\n" );
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document.write( "Differentiate it by [with respect to] the time:\r\n" );
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document.write( " 
= 
. (2)\r\n" );
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document.write( " is given: it is the inflow rate, or 8 m^3/hour. \r\n" );
document.write( "Hence, when h = 3 meters deep, we have from (2)\r\n" );
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document.write( "Notice that she waited until after differentiating the equation for V, to\r\n" );
document.write( "substitute the \"freeze value\" of 3 for the variable h. Never substitute the\r\n" );
document.write( "\"freeze value\" of a variable too early, but wait until after the calculus has\r\n" );
document.write( "been applied to substitute it. \r\n" );
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document.write( " 8 = 
= 
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document.write( "Hence, 
= 
= 0.78595 meters per hour.\r\n" );
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document.write( "Rounding to 3 decimals, you get the \r\n" );
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document.write( "ANSWER. When the water depth is 3 meters, the water rising rate is about 0.786 m/hour.\r\n" );
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document.write( "Notice that the water rising rate of 0.786 m/h occurred only at one instant!\r\n" );
document.write( "One second before the water became 3 inches deep, the water rising rate was\r\n" );
document.write( "slightly faster, and then, one second later, the water rising rate was slightly\r\n" );
document.write( "slower. The water rising rate continued to get slower and slower until the\r\n" );
document.write( "cone became completely filled. This is because the top of the cone of water\r\n" );
document.write( "became wider and wider as more water was added. The water rising rate would be\r\n" );
document.write( "constant when filling a cylinder, but not when filling a cone. \r\n" );
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