Calculate and interpret average and instantaneous rates from a model

Construct the geometric model before adding time

Write cone and similar-cylinder volumes as functions of one length before differentiating. A derivative of volume with respect to depth has units of volume per length; a time rate also needs the depth's rate of change with time. State the geometric constraint before using it

A fixed cone gives a similarity relation

For a fixed cone of radiusR=6cm and heightH=9cm, a horizontal water surface gives r/h=2/3. Thus V=πr²h/3=(4π/27)h³ and dV/dh=(4π/9)h². At water depth3cm, dV/dh=4πcm². The depth is positive and below the rim

A cone has radius 6 cm and height 9 cm. At a water depth of 3 cm, give the coefficient of π in the derivative of volume with respect to depth

A family of similar cylinders is a different assumption

If a family of cylinders satisfies h/r=3/2 throughout, then V=πr²h=(3π/2)r³ and dV/dr=(9π/2)r². At r=4m the derivative is72πm². This is a changing family with a fixed ratio; it is not a fixed cylindrical tank whose height changes independently of radius

A family of similar cylinders has a fixed height-to-radius ratio of 3/2. At a radius of 4 m, give the coefficient of π in the derivative of volume with respect to radius

Now connect a geometric derivative to time

In the fixed-cone example, suppose the water depth is falling at1/5cm/s when h=3cm. Then dV/dt=(dV/dh)(dh/dt)=4π×(−1/5)=−4π/5cm³/s. Differentiate the relationship first; only then substitute this instant’s depth and signed rate

A cone has radius 6 cm and height 9 cm. At a water depth of 3 cm, the depth is falling at 1/5 cm per second. Give the coefficient of π in the signed volume rate

Volume is measured in cm³, depth in cm and time in seconds. Which units result from multiplying the volume change per unit depth by the depth change per unit time?

Distinguish length and time rates

A volume model and its derivative with respect to length do not by themselves give a time rate. Use a time-dependent chain for linked dimensions, or retain both terms of the product rule when radius and height change independently