#### 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.