Problem 67 - College Algebra

Question

Use the written statements to construct a polynomial function that represents the required information. A cube has an edge of 3 feet. The edge is increasing at the rate of 2 feet per minute. Express the volume of the cube as a function of m, the number of minutes elapsed.

Step-by-Step Solution

Answer

The polynomial function representing the cube's volume is \( V(m) = 8m^3 + 36m^2 + 54m + 27 \).

Step 1: Understand the given information

We know that a cube has an edge of 3 feet initially, and this edge is increasing at a rate of 2 feet per minute. We also want to express the volume as a function of time, represented by 'm', which is the number of minutes elapsed.

Step 2: Define the edge length as a function of time

Given that the edge is initially 3 feet and increases at 2 feet per minute, we can express the edge length formula as: \[ e(m) = 3 + 2m \] where \( e(m) \) represents the edge length in feet, and \( m \) is the time in minutes.

Step 3: Write the formula for the volume of a cube

The volume of a cube is calculated by cubing its edge length. Therefore, the volume \( V \) as a function of the edge, \( e \), is:\[ V(e) = e^3 \]

Step 4: Substitute the edge length into the volume equation

Substitute the expression for the edge length \( e(m) = 3 + 2m \) into the volume formula:\[ V(m) = (3 + 2m)^3 \]

Step 5: Expand the polynomial

Expand the expression \((3 + 2m)^3\) using the binomial theorem or algebraic expansion:\[ V(m) = (3 + 2m)(3 + 2m)(3 + 2m) \]Carrying out the multiplication:1. \((3 + 2m)(3 + 2m) = 9 + 12m + 4m^2\)2. \((9 + 12m + 4m^2)(3 + 2m) = 27 + 54m + 36m^2 + 8m^3\)Combining like terms gives us:\[ V(m) = 27 + 54m + 36m^2 + 8m^3 \]

Step 6: Write the final polynomial function

The volume of the cube as a function of the number of minutes elapsed is given by the polynomial:\[ V(m) = 8m^3 + 36m^2 + 54m + 27 \]

Key Concepts

Cube Volume

When we think about cubes, the first thing that comes to mind is that they are three-dimensional shapes with equal sides. This simple feature means that if you know the length of one side, you can find the volume of the cube.
The formula to calculate the cube's volume is \[ V = e^3 \]where \( V \) is the volume and \( e \) is the edge length.
In our exercise, the cube starts with an edge of 3 feet, so initially, the volume equals \( 3^3 = 27 \) cubic feet.
This initial value grows as the edge length increases over time.

Polynomial Expansion

Polynomial expansion is a technique used to transform expressions like \((a + b)^n\) into expanded form. This makes calculations easier and expressions clearer. In our exercise, we use this technique on the expression \( (3 + 2m)^3 \).
The binomial theorem, or simply algebraic manipulation, helps us break this down into simpler terms. It involves multiplying out the binomial terms step-by-step:

• First, calculate \( (3 + 2m)(3 + 2m) = 9 + 12m + 4m^2 \).
• Then, use the result \( (9 + 12m + 4m^2) \) and multiply it by another \( (3 + 2m) \), leading to \( 27 + 54m + 36m^2 + 8m^3 \).

This expansion results in a polynomial, with decreasing powers of \( m \), that expresses how volume changes with time.

Rate of Change in Geometry

Rate of change in geometry helps us understand how a specific shape's dimension changes over time. For a cube, if its edge length alters at a constant rate, the volume will change accordingly.
In our problem, the cube's edge increases at 2 feet per minute. This steady increase turns the edge length at time \( m \) into \( e(m) = 3 + 2m \).
With each passing minute, a small change in edge length results in a significant change in cube volume, because volume depends on the cube of the edge.
Calculating the derivative of our polynomial \[ V(m) = 8m^3 + 36m^2 + 54m + 27 \] can find us the exact rate at which the volume is changing at any given minute.