   Chapter 2.P, Problem 26P

Chapter
Section
Textbook Problem

# A cone of radius r centimeters and height h centimeters is lowered point first at a rate of 1cm/s into a tall cylinder of radius R centimeters that is partially filled with water. How fast is the water level rising at the instant the cone is completely submerged?

To determine

To find:

the rate at which water level rising at the instant the cone is completely submerged dx/dt.

Solution:the rate at which water level rising at the instant the cone is completely submerged is r2R2-r2 cm/s

Explanation

1) Formula:

i. Volume of cone

V=πr2.h3

ii. Volume of cylinder

V=πr2h

2) Given:

i. r=radius of cone in cm

ii. h=height  of cone in cm

iii. H=initial height of water

iv. R=radius of cylinder

3) Calculation:

Assume the axes of the cone and the cylinder are parallel.

Let H denote the initial height of the water.

When the cone has been dropping for t seconds, the water level has risen x centimeters, so the tip of the cone is x+1t cm below the water line.

To find dx/dt when x+t=h( when the cone is completely submerged).

Using the similar triangle,

r1x+t=rh

r1=rh(x+t)

volume of water and cone at time t=original volume of water+volume of submerged part of cone

πR2H+x=πR2H+13πr12(x+t)

πR2H+πR2x=πR2H+13πr2<

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