Concepts addressed: cylinder volume, cone volume, unit conversion.
Recommended grade: 9th.
Difficulty level: Advanced.
A basic version of Hair volume is also available.
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.
- 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). How many average heads of 3-inch-long hair could you fit in a one-gallon bottle?
- A man’s facial hairs tend to be thicker than their head hairs. Assume that the average beard hair is 0.004 in thick, and let’s think about our beloved hairy half giant – Hagrid. Let’s say that he has 150,000 hairs on his head and 80,000 on his beard, and that both are 9-inch-long. If every cubic inch of hair needs 0.07 fl oz of shampoo and a 12 fl oz bottle of shampoo costs $15, how much does he spend for that alone in the course of a week? Obviously, Hagrid washes his magnificent hair and beard every single day. Also, assume for simplicity that his hair doesn’t grow at all.
- 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:
Now we’re ready to count the volumes. Let’s start with the cylinder and the cylinder volume calculator:
cylinder = h_cylinder * π * r² = 2 in * π * (0.0015 in)² ≈ 0.0000141 in³,
and now the cone with the right circular cone calculator:
cone = (h_cone * π * r²) / 3 = (1 in * π * (0.0015 in)²) / 3 ≈ 0.0000024 in³.
This means that a single strand of hair’s volume is
strand = cylinder + cone ≈ 0.0000165 in³.
Therefore, the volume of all 100,000 of those is:
V = 100,000 * strand ≈ 1.65 in³.
We know that one US gallon is 231 cubic inches. If we divide that by the volume of a single head of hair, we’ll get
231 in³ / 1.65 in³ = 140.
So our one-gallon bottle can fit 140 heads of hair.
Let’s begin with the head hair. Again, we divide it into a cylinder and a cone, the first will be of height h_Hcyliner = 6 in (two-thirds of the whole 9 inches of length) and the second of height h_Hcone = 3 in.
Now we calculate the volume of each part using the cylinder volume calculator and the right circular cone calculator:
head_cylinder = h_Hcylinder * π * r² = 6 in * π * (0.0015 in)² ≈ 0.0000424 in³,
head_cone = (h_Hcone * π * r²) / 3 = (3 in * π * (0.0015 in)²) / 3 ≈ 0.0000071 in³,
and so the volume of a single head hair is
head_strand = head_cylinder + head_cone ≈ 0.0000495 in³.
Lastly, the volume of all 150,000 of Hagrid’s head hair is
head_V = 150,000 * head_strand ≈ 7.43 in³.
NOTE: We could have also observed that a single hair in question 2 is three times as long as that from question 1, and therefore the cylinder and the cone part’s heights are also three times what they were before. This means that the volume of a single hair is three times as much as what we obtained in question 1. Note, however, that Hagrid has many more strands of hair on his head than a regular person.
Now we repeat the calculations for the beard hair. The length of the hair is the same, so the heights of the cylinder and the cone are again h_Bcylinder = 6 in and h_Bcone = 3 in respectively. However, the diameter of the base of each of them is now 0.004 in, so the radius is half of that, which is R = 0.002 in.
Let’s get back to the cylinder volume calculator and the right circular cone calculator to obtain
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 in³.
Lastly, the volume of all 80,000 of Hagrid’s beard hair is
beard_V = 80,000 * beard_strand ≈ 7.04 in³.
This means that all in all, Hagrid has 7.43 in³ + 7.04 in³ = 14.47 in³ of hair to wash every day. Since each cubic inch needs 0.07 fl oz of shampoo, he will need
14.47 * 0.07 fl oz ≈ 1.01 fl oz
of shampoo every day. Weekly, this gives
1.01 fl oz * 7 = 7.07 fl oz.
Therefore, his hair care costs him
(7.07 / 12) * $15 ≈ $8.84