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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

A cone of radius r centimeters and height h centimeters is lowered point first at a rate of 1 cm/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 the water level is rising at the instant the cone is completely submerged.

Explanation

Assume that x is the cone’s lowered height in water and y is the height at which the water came up.

The rate of fall of cone in water is 1 cm/s. That is, dxdtdydt=1.

The cone is submerged at a height x.

The submerged volume of the cone is Vdown=13πr20x where r0 is the radius.

From Figure 1, tanθ=rh=r0x

rh=r0xr0=rhx

The volume of water rising in the cylinder due to the fall of cone is Vup=(πR2πr20)y

The volume of the cone submerged is equal to the volume of water.

That is,

Vdown=Vup13πr02x=(πR2πr02)y

Substitute r0=rhx in the above equation,

13π(rhx)2x=(πR2π(rhx)2)y13π(rh)2x3=(πR2π(rh)2x2)y

Since R>r>r0, the second term in the right hand side of the above equation can be neglected.

13π(rh)2x3=(πR2)y

Differentiate on both sides with respect to t

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