A conical paper cup is 10 cm tall with a radius of 10 cm. The cup is being filled with water so that the water level rises at a rate of 2 cm/sec. At what rate is water being poured into the cup when the water level is 8 cm?

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Answer 1 · 2016-08-05T09:31:59

# = 32 pi \ cm^3 "/ sec"#

the volume of a cone is #V=( pi r^2 h)/3#

in this case, we can relate #h# and #r# as we know that the slope = #h/r = 10/5 = 2#

we can make the volume a formula in a single variable

#V=( pi (h/2)^2 h)/3 = ( pi h^3)/12#

differentiating wrt time, it follows that

#dot V = (3 pi h^2 dot h)/12= ( pi h^2 dot h)/4#

so to address the question

#dot V = ( pi 8^2 * 2)/4 = 32 pi \ cm^3 "/ sec"#