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Chapter 1: Problem 27

Estimate the mass of your head. Assume that its density is that of water, \(1000 \mathrm{~kg} / \mathrm{m}^{3}\)

Short Answer

Expert verified
Based on the assumption that the density of a human head is the same as water, 1000 kg/m³, and the average adult head dimensions, the estimated mass of a human head is approximately 2.64 kg. Keep in mind that this estimation is based on an ellipsoid model and may vary for each individual.

Step by step solution

01

Approximate the volume of your head

To make this estimation, you can use the average adult head dimensions found in various studies. The human head can be roughly modeled as an ellipsoid. The general formula for the volume of an ellipsoid is given by: \(V = \frac{4}{3}\pi abc\) Where \(a\), \(b\), and \(c\) are the semi-axes lengths of the ellipsoid. You can find reference measurements for an average head and use those as the semi-axis lengths. For example, using the measurements: \(a = 9.5\mathrm{~cm}\), \(b = 8\mathrm{~cm}\), and \(c = 11.5\mathrm{~cm}\) Based on these approximations, let us calculate the volume.
02

Calculate the volume using the ellipsoid formula

Plugging the dimensions into the ellipsoid formula, you get: \(V = \frac{4}{3}\pi (9.5\mathrm{~cm})(8\mathrm{~cm})(11.5\mathrm{~cm})\) Now, calculate the volume: \(V \approx 2636.1 \mathrm{~cm}^3\)
03

Convert volume to m³

The volume calculated above is in cubic centimeters. Since the density given is in kg/m³, we will need to convert the volume to cubic meters, as follows: \(V \approx \frac{2,636.1\mathrm{~cm^3}}{100^3}\mathrm{~m^3}\) \(V \approx 2.6361 \times 10^{-5}\mathrm{~m^3}\)
04

Compute the mass using the density formula

The mass of your head can be computed using the formula for density, which is given by: \(\rho = \frac{m}{V}\) Rearranging the formula to solve for mass, we get: \(m =\rho \times V \) Now, plug in the density of water and the calculated volume of your head: \(m =(1000\mathrm{~kg/m^3})(2.6361\times 10^{-5} \mathrm{~m^3})\) Calculate the mass: \(m\approx 2.6361\mathrm{~kg}\) So, the estimated mass of your head is approximately 2.64 kg. Remember, this is just an estimation based on assumptions and the actual mass may be different for every individual.

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