To determine

(a)

To guess:

Which napkin rings has more wood in it.

Explanation

1) Concept:

If the volume of an object is more, then it requires more material.

2) Calculation:

From the figure see that,

 The second napkin ring may have more volume than the first napkin ring.

So, it requires more wood than the first napkin ring.

Conclusion:

The second napkin ring has more wood.

To determine

(b)

To check:

Your guess

Answer

Our guess is incorrect. The volume of both the rings is the same.

Explanation

1) Concept:

i. If x is the radius of the typical shell, then the circumference =2πx and the height is y

ii. By the shell method, the volume of the solid by rotating the region under the curve y=f(x) about y- axis from a to b is

V= ∫ab2πxf(x)dx

where,  0≤a≤b

2) Given:

The napkin rings have the same height h.

3) Calculation:

The napkin rings are as shown in figure.

From the figure, see that the napkin rings has a spherical shape.

Let r be the radius of the hole and R  be the radius of the sphere.

Therefore, the napkin ring is obtained by drilling a hole of the radius r through a sphere of the radius R.

That is the napkin ring is obtained by rotating the region bounded by the curves x2+y2=R2, x=r about y-axis

Solving for y we have ⇒y= ±R2-x2 and x=r, x=R

Using the shell method, find the typical approximating shell with radius x,

Therefore, the circumference is 2πx and the height is y= R2-x2--R2-x2=2R2-x2

So, the total volume is

V= ∫ab2πx 2R2-x2dx The integral is from r to R.

V=- 2π∫rR(-2x)R2-x212dx

Let -x2=t⇒-2x dx=dt

x=R⇒t=-R2 and x=r⇒t=-r2

V=- 2π∫-r2-R2R2+t12dt

V=-2π2R2+t323-r2-R2

=-2π 2R2-R2323-2R2-r2323

=2π2R2-r2323

V=4πR2-r2323

By the Pythagorean Theorem,

R2-r2=(12h)2(refer to the figure above)

Therefore,

V=4π(12h)2323=4π12h33

V=πh36

Therefore, the volume of the napkin ring is V=πh36unit3

Volume is independent of R and r it is depends only on the height h and the height of both the napkin rings are the same.

Therefore, the amount of the wood is the same for both the napkin rings.

Therefore, our guess is incorrect.

Conclusion:

 Our guess is incorrect.