Concepts addressed: **cylinder volume**, **cone volume**.

Recommended grade: **9th**

Difficulty level: **Basic**.

An advanced version of Hair volume is also available.

**Scenario:**

Every shampoo advert tell us that their product will boost our hair’s volume, but have you ever wondered what** the volume of our head of hair** actually is? We know, right, who hasn’t?! The average hair is about **0.003 in thick**. The average human head has **about 100,000 hair follicles**.

- On average, a human hair grow about
**0.5 inches every month**. How much additional volume will a person grow in half a year, if they started off bald? For simplicity, let’s say that**two-thirds of a hair’s length is a cylinder**, while**the remaining length is a right circular cone**(see the diagram below). - A man’s facial hairs tend to be thicker than their head hairs. If we assume that
**the average beard hair is 0.004 in thick**and the average beard**consists of 30,000 such hairs**, what takes up more volume: Gandalf’s stylish**6-inch-long hairdo**or his magnificent**9-inch-long wizardly beard**?

**Useful calculators:**

- Cylinder volume calculator – https://www.omnicalculator.com/math/cylinder-volume
- Right circular cone calculator – https://www.omnicalculator.com/math/right-circular-cone
- Conversion calculator – https://www.omnicalculator.com/conversion/conversion-calculator

**Question 1 hints:**

**Question 2 hints:**

**Solutions:**

**Step-by-step solution:**

**0.5 inches every month**, then after half a year it will be 6 * 0.5 in = 3 in long. Let’s begin by calculating

**the volume of a single strand of such a hair.**

The description tells us that one part of it is a cylinder, and the other part is a cone. Also, the cone makes up a third of the length of a hair, and we want the hair to be 3-inch-long, so the **cone’s height** will be h_cone = 3 in / 3 = 1 in. This means that **the cylinder’s** **height** will be h_cylinder = 2 in. Moreover, a hair is 0.003 in thick, so this will be the diameter of the base of both the cylinder and the cone. Therefore, **the radius of each of them** will be half of that, which is r = 0.0015 in.

Now we’re ready to calculate the volumes. Let’s start with **the cylinder**, using the cylinder volume calculator:*cylinder = h_cylinder * π * r² = 2 in * π * (0.0015 in)² ≈ 0.0000141 in³*,

Next is

**the cone**and the right circular cone calculator:

*cone = (*

*h_cone ***π * r²*) / 3 =*(1 in **.

*π * (0.0015 in)²*) / 3*≈*0.0000024 in³This means that

**the**

**volume of a single strand of hair**is

*strand = cylinder + cone*.

*0.0000165 in**≈**³*The last thing to do is to calculate **the volume of all 100,000** **hairs**:*V = 100,000 * strand ≈ 1.65 in³*

**the volume of a single strand of hair**. Note, however, that we have to do it

**separately for head hair and beard hair**since they have different thicknesses and lengths.

We’ll begin with **the head hair**. Again, we divide it into a cylinder and a cone, where the cylinder’s height, h_Hcylinder, is 4 in (two-thirds of the whole 6 inches of length), and the cone’s height, h_Hcone, is 2 in.

Now we move on to finding the volume of each shape by using the cylinder volume calculator and the right circular cone calculator respectively:*head_cylinder = h_Hcylinder * π * r² = 4 in * π * (0.0015 in)² ≈ 0.0000282 in³*,

*head_cone =*=

*(**h_Hcone * π * r²) / 3**.*

*(2 in ***π * (0.0015 in)²*) / 3*≈*0.0000047 in³Adding these two results together gives that

**the volume of a single head hair**, which is:

*head_strand = head_cylinder + head_cone*.

*≈*0.0000329 in*³*Lastly, the volume of **all 100,000 of Gandalf’s head hair** is*head_V = 100,000 * head_strand ≈ 3.29 in³*.

**NOTE**: We could have also observed that a single hair in Question 2 is twice as long as that from Question 1, and therefore the cylinder and the cone parts’ heights are also twice what they were before. This means that the volume of a single hair, and so also of all the head hairs is twice as much as what we obtained in Question 1. Note, however, that this gives a small difference of 0.01 in³ in the final answer because of rounding.

Now we repeat the calculations for** the beard hair**. Here the length of the hair is 9 in, which means that the cylinder and the cone have **heights** h_Bcylinder = 6 in and h_Bcone = 3 in respectively. Moreover, the diameter of the base of such a hair changes to 0.004 in, so **the radius** is now half of that, which is R = 0.002 in.

Let’s get back to the cylinder volume calculator and the right circular cone calculator and compute*beard_cylinder = h_Bcylinder * π * R² = 6 in * π * (0.002 in)² ≈ 0.0000754 in³*,

*beard_cone =*

*(**h_Bcone * π * R²) / 3**=*,

*(3 in * π * (0.002 in)²) / 3 ≈ 0.0000126 in³*and so

**the volume of a single beard hair**is

*beard_strand = beard_cylinder + beard_cone*.

*≈*0.000088 in*³*Lastly, **the volume of all 30,000 of Gandalf’s beard hair** is*beard_V = 30,000 * beard_strand ≈ 2.64 in³*.

In particular, we see that the hairdo has more volume than the beard.

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