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Chapter 1: Problem 23

Assume a dog's brain is twice as great in diameter as a cat's, but each animal's brain cells are the same size and their brains are the same shape. In addition to being a far better companion and much nicer to come home to, how many times more brain cells does a dog have than a cat? The answer is not \(2 .\)

Short Answer

Expert verified
A dog has eight times more brain cells than a cat.

Step by step solution

01

Understanding the Problem

We need to determine the ratio of brain cells between a dog and a cat, given that they have the same brain cell size and shape, but the dog's brain's diameter is twice that of the cat's.
02

Volume Relationship

Since both brains are the same shape, we assume they are spheres. The volume of a sphere is given by the formula \(V = \frac{4}{3}\pi r^3\), where \(r\) is the radius of the sphere.
03

Relationship of Diameters to Radius

If the diameter of the cat's brain is \(d\), then its radius \(r_c = \frac{d}{2}\). The dog's brain has a diameter of \(2d\), so its radius \(r_d = d\).
04

Calculate Volumes

Using the formula for volume, the volume of the cat's brain is \(V_c = \frac{4}{3} \pi \left(\frac{d}{2}\right)^3 = \frac{4}{3} \pi \frac{d^3}{8} = \frac{\pi d^3}{6}\). The volume of the dog's brain is \(V_d = \frac{4}{3} \pi d^3\).
05

Find the Ratio

To find how many times more brain cells the dog has, divide the dog's brain volume by the cat's brain volume: \(\frac{V_d}{V_c} = \frac{\frac{4}{3} \pi d^3}{\frac{\pi d^3}{6}} = \frac{\frac{4}{3}}{\frac{1}{6}} = 8\).
06

Conclude the Result

Since they have the same brain cell density, the dog has eight times more brain cells than the cat, because the brain volume is eight times larger.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sphere
In mathematics and geometry, a sphere is a perfectly round three-dimensional shape. It's like a 3D circle. A sphere has no edges or vertices and all points on the surface are equidistant from a single point in the middle.

Think of a basketball or a bubble鈥攖hese are great examples of real-life spheres.

When it comes to calculations, especially in problems involving volume, understanding the spherical shape is essential. For example, when considering a brain shaped like a sphere, you can calculate its volume with ease using the sphere volume formula, which we'll discuss later. This method works because the brain is modeled as a geometrically simple and regular shape, making computations straightforward.
Brain cells
Brain cells, or neurons, are the working units of the brain responsible for carrying out all the functions of this complex organ. Despite their small individual size, when packed together in vast numbers, they form the brain's volume.

In animals like dogs and cats, while the size of individual brain cells can be similar, the total number varies with the size of the brain. This is why, in our original exercise, we focus on the brain's size to determine the difference in the number of brain cells between animals.

Given that the dog's and cat's brain cells are of the same size, the difference primarily comes from the volume of their brains. A larger brain can house more brain cells, provided the density (the number of cells per unit of volume) remains constant.
Ratio
A ratio is a relationship between two numbers that indicates how many times the first number contains the second. Ratios can be extremely useful in comparing quantities and understanding proportional relationships.

In the context of the exercise, we use the ratio to compare the number of brain cells in a dog's brain to those in a cat's brain. By knowing the size and volume of each brain, we can determine the change in number by calculating the ratio.

As it turns out, the dog's brain volume, being eight times larger, implies a ratio of 8:1. This means for every brain cell in the cat's brain, there are eight in the dog's brain, assuming the density of brain cells is consistent.
Volume formula
The formula to calculate the volume of a sphere is \[ V = \frac{4}{3}\pi r^3 \]where "\(V\)" represents volume, "\(r\)" is the radius, and \(\pi\) is a constant approximately equal to 3.14159. This formula is central to solving problems related to spherical shapes, like determining how many brain cells fit into a brain modeled as a sphere.

In the original problem, knowing the diameter of the brains helps us find their radii. For the cat's brain, if the diameter is \(d\), the radius becomes \(\frac{d}{2}\), whereas for the dog's brain, with its diameter being \(2d\), the radius is \(d\).

This difference in radius affects the calculated volume. Since volume depends on the cube of the radius, even a slight increase in radius leads to a substantial increase in volume and thus the number of brain cells. Hence, using the volume formula is crucial for understanding and quantifying these differences.

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Most popular questions from this chapter

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