3D geometry – Hair volume (advanced)

Concepts addressed: cylinder volume, cone volume, unit conversion.
A basic 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.

1. 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?
2. 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.

Useful calculators:

Question 1 hints:

Hint 1
Try to sketch what shape a single hair is. What does it consist of?
Hint 2
What is the height of the cylinder part of one hair? And of the cone part?
Hint 3
What is the volume of a single hair? Remember that in calculations, we need all the parameters to be expressed in the same units.
Hint 4
Knowing the volume of a single hair, how much volume does a whole head of hair correspond to? How does that translate to one-gallon bottles? Again, remember that you need compatible units.

Question 2 hints:

Hint 1
We know from the first question how to calculate the volume of a single hair. What is it with our current parameters? Remember to consider head hair and beard hair separately.
Hint 2
If we know the volumes of a single head hair and a single beard hair, how much hair volume does the giant’s head have? And the beard?
Hint 3
Knowing how much volume of hair Hagrid has, how much shampoo does he use every time? And how much does he need in a week in total? How does that translate to the price of it?

Solutions:

Question 1
Question 2
cost ≈ \$8.84

Step-by-step solution:

Question 1
Let’s begin by calculating the volume of a single strand of hair. The description tells us that one part of it is a cylinder and another is a cone. Also, the cone is 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 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 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.

Question 2
Similarly to what we did in the first question, we’ll start by calculating 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 thickness.

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
Lastly, the volume of all 150,000 of Hagrid’s head hair is

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
each week.

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