Problem 19 - Calculus Early Transcendentals: Pearson New International Edition

Question

A wire of length 8 centimeters is such that the mass between its left end and a point \(x\) centimeters to the right is \(x^{3}\) grams (Figure 12). Figure 12 (a) What is the average density of the middle 2 -centimeter segment of this wire? Note: Average density equals mass/length. (b) What is the actual density at the point 3 centimeters from the left end?

Step-by-Step Solution

Answer

The average density of the middle segment is 49 g/cm and the actual density at 3 cm is 27 g/cm.

Step 1: Define the Problem

We need to find the average density of the middle 2-centimeter section and the actual density at a point 3 centimeters from the left end of a wire. The mass of the wire segment between its left end and a point at a distance \( x \) centimeters is given by \( x^3 \) grams.

Step 2: Identify Middle Segment

The middle 2-centimeter segment of the wire is between \( x = 3 \) and \( x = 5 \) centimeters. We need to calculate the mass within this part of the wire.

Step 3: Calculate Mass of Middle Segment

To find the mass of the segment from 3 to 5 centimeters, compute: \[ \text{Mass from } x = 3 \text{ to } x = 5 = 5^3 - 3^3 = 125 - 27 = 98 \text{ grams}. \]

Step 4: Calculate Average Density of Middle Segment

The length of the middle segment is 2 centimeters. Therefore, the average density is given by \( \text{Average Density} = \frac{\text{Mass}}{\text{Length}} = \frac{98 \text{ grams}}{2 \text{ cm}} = 49 \text{ g/cm}. \)

Step 5: Determine Actual Density at x=3

The density at a point \( x \) is the derivative of the mass function \( x^3 \) with respect to \( x \). Compute: \[ \rho(x) = \frac{d}{dx}(x^3) = 3x^2. \] Substitute \( x = 3 \) to find the density at that point: \( \rho(3) = 3(3)^2 = 27 \text{ g/cm}. \)

Key Concepts

Mass Calculations

Calculating the mass of an object involves determining how much matter it contains. In mathematical problems involving varying density, this can be intricate. Here, the mass of the wire segment is calculated using the given mass function \(x^3\), where \(x\) is the distance from the wire's left end.
To find the mass of a specific segment, subtract the mass at the left endpoint from the mass at the right endpoint. For example, if you wish to find the mass of the wire from \( x=3 \) centimeters to \( x=5 \) centimeters, evaluate the mass function at both endpoints and subtract:

• Mass at \(x = 5\): \(5^3 = 125\) grams
• Mass at \(x = 3\): \(3^3 = 27\) grams
• Mass of the segment: \(125 - 27 = 98\) grams

This process helps determine how much mass is concentrated over a given part of the wire.

Density Functions

Density is a measure of how much mass is concentrated in a given volume or length. It's a measure of 'compactness' and is commonly used in physics to describe material properties.
In the context of the exercise, we find the average density of a segment and the actual density at a specific point. The average density is the total mass divided by the length of the segment. This provides a general idea of the mass distribution over that length.
For the middle 2-centimeter segment, the average density calculation is straightforward:
\[\text{Average Density} = \frac{\text{Mass}}{\text{Length}} = \frac{98 \text{ grams}}{2 \text{ cm}} = 49 \text{ g/cm}\]
This tells us that, on average, 49 grams of wire mass occupy each centimeter of this specific segment.

Derivative Applications

In mathematical problems, derivatives help determine rates of change. When dealing with a function that represents mass, the derivative can show how density varies across the wire.
The problem provides a mass function \( x^3 \) grams for a distance \( x \). To find the density at any point \( x \), we calculate the derivative of this function, which gives us \( 3x^2 \).
By evaluating this derivative at \( x = 3 \), we find the actual density at that point on the wire:
\[\rho(x) = \frac{d}{dx}(x^3) = 3x^2\]
Plugging in \( x = 3 \), we have:
\[\rho(3) = 3(3)^2 = 27 \text{ g/cm}\]
This calculation indicates how densely mass is packed at precisely 3 centimeters from the wire's starting point.

Mathematical Problem Solving

Solving mathematical problems involves identifying what is given and applying appropriate methods to find a solution. This exercise demonstrates problem-solving through sequential steps.
Initially, we identify the section of the wire and its mass capabilities. The goal is clear: calculate average and actual density. Using functions, derivatives, and logical thinking, we break down complex problems into simpler parts:

• Define the problem and identify relevant points (e.g., endpoints of the wire segment)
• Use mathematical operations (like subtraction and division) to calculate mass first, then density
• Apply calculus concepts (such as derivatives) to model and solve intricate parts of the problem

These steps not only solve the immediate problem but also build a foundation for handling similar calculations in future scenarios.