To determine

a)

Volume V as a function of height h

Answer

V=∫0hπfy2 dy

Explanation

1) Concept:

Let S be a solid that lies between y=a  and y=b. If the cross sectional area of S in the plane Py through y and perpendicular to the y axis is A(y), where A  is a continuous function, then volume of S is

V=∫abAydy

If the cross section formed is the disk, then find radius of the disk in terms of y and use

Ay=πradius2

2) Given:

x=f(y)

3) Calculations:

It is given that the container is formed by rotating the graph of f about y axis.

Consider the graph as shown below.

When sliced through y, it forms the shape of a container.

So, the area of cross section  through y(which is a disk) is

Ay=πx2=πfy2

Since the volume is to be calculated for height h, integrate from 0 to h. Therefore,

V(y)=∫0hAydy

V(y)=∫0hπfy2dy

Conclusion:

V(y)=∫0hπfy2dy

To determine

(b)

To prove:

dVdt=πfh2·dhdt

Answer

dVdt=πfh2·dhdt

Explanation

1) Concept:

i) First Fundamental Theorem of calculus:

If f is continuous on [a, b], then the function g defined by

gx=∫axf(t)dt

is continuous on a, b and differentiable on a, b and g'x=fx

ii) Chain Rule:

If y=f(u) and u=gx are both differentiable functions, then

dydx=dydu·dudx

2) Calculation:

From the previous calculation,

V=∫0hπfy2dy

Differentiating with respect to t, by using Chain rule,

dVdt=dVdh·dhdt

dVdt=ddh∫0hπfy2dy·dhdt

Then by fundamental theorem of calculus,

dVdt=πfh2·dhdt

Hence the given equation is proved.

Conclusion:

dVdt=πfh2dhdt

To determine

(c)

The formula for the function f and write the advantage of having dhdt constant.

Answer

fh=kAπCh14

The advantage of taking  dhdt=C is that the markings on the container are equally spaced.

Explanation

1) Concept:

Use the equation for dVdt from part a, and use the given information in it. Then solve for fh.

2) Given:

a)

dVdt=kAh

b)

dhdt=C

3) Calculation:

From the part a,

dVdt=πfh2dhdt

Substituting the given values,

kAh=πfh2C

Simplifying,

fh2=kAhπC

Taking square root,

fh=kAπC(h)14

The advantage of taking dhdt=C is that there will be a linear change between h and t. Therefore, the change in height h  will vary linearly with respect to the change in time t.

So, the advantage of taking  dhdt=C is that the markings on the container are equally spaced.

Conclusion:

fh=kAπC(h)14

The advantage of taking  dhdt=C is that the markings on the container are equally spaced.