Problem 73 - College Algebra

Question

For the following exercises, find the dimensions of the box described. The length is one inch more than the width, which is one inch more than the height. The volume is 86.625 cubic inches.

Step-by-Step Solution

Answer

The dimensions are: Height = 3.5 inches, Width = 4.5 inches, Length = 5.5 inches.

Step 1: Understand Relationships Between Dimensions

Let's denote the height of the box as \( h \). We are told that the width is one inch more than the height, so the width \( w \) is \( h + 1 \). The length is one inch more than the width, so the length \( l \) is \( h + 2 \). Thus, the dimensions of the box are expressed in terms of the height: \( l = h + 2 \), \( w = h + 1 \), and \( h = h \).

Step 2: Write the Volume Equation

The volume of the box can be represented as the product of its length, width, and height: \( V = l \cdot w \cdot h \). Substituting the expressions for \( l \), \( w \), and \( h \), we obtain the equation: \( V = (h + 2)(h + 1)(h) = 86.625 \).

Step 3: Expand and Simplify the Volume Equation

First, expand the equation: \((h + 2)(h + 1) = h^2 + 3h + 2\). Then multiply by \( h \): \( h(h^2 + 3h + 2) = h^3 + 3h^2 + 2h \). So, \( h^3 + 3h^2 + 2h = 86.625 \).

Step 4: Solve for the Height

We have the cubic equation \( h^3 + 3h^2 + 2h = 86.625 \). By trial or using numerical methods, substitute values for \( h \) to find the one that satisfies the equation. Trying \( h = 3.5 \), we compute \( 3.5^3 + 3(3.5)^2 + 2(3.5) = 42.875 + 36.75 + 7 = 86.625 \). Therefore, \( h = 3.5 \) is the correct height.

Step 5: Calculate Width and Length

Now calculate the width using \( w = h + 1 \): \( w = 3.5 + 1 = 4.5 \). Similarly, calculate the length using \( l = h + 2 \): \( l = 3.5 + 2 = 5.5 \). Hence, the dimensions of the box are: height = 3.5 inches, width = 4.5 inches, length = 5.5 inches.

Key Concepts

box dimensions

In this problem, we're finding the dimensions of a box with a given volume of 86.625 cubic inches. The relationship between the dimensions is key to solving it. The dimensions of a box are commonly referred to as length, width, and height, symbolized as \( l \), \( w \), and \( h \), respectively. For this specific box, there is a clear relationship that ties these three dimensions together:

• Height \( h \) is the starting point.
• Width \( w \) is one inch more than the height, so \( w = h + 1 \).
• Length \( l \) is one inch more than the width, hence \( l = h + 2 \).

These relationships help us create expressions for each dimension, which can then be used in solving the problem. This method of defining dimensions in relation to one variable simplifies the mathematical process considerably, making it easier to plug values into the volume formula.

cubic equations

Cubic equations are polynomials of degree three, typically taking the form \( ax^3 + bx^2 + cx + d = 0 \). In our exercise, a cubic equation arises when determining the height of the box. Given the volume formula \( V = l \cdot w \cdot h \), we substitute our expressions for \( l \), \( w \), and \( h \) into it, leading to the equation:\[ (h + 2)(h + 1)(h) = 86.625 \]When expanded, this results in:\[ h^3 + 3h^2 + 2h = 86.625 \]This is a cubic equation. To solve it, various methods can be used, including:

• Trial and Error: Testing different values of \( h \) to see if they satisfy the equation.
• Numerical Methods: Using calculators or software to estimate roots.

For this exercise, testing \( h = 3.5 \) provides a solution that satisfies the equation. Solving cubic equations often involves a bit of iteration, especially when exact solutions aren't obvious.

algebraic expressions

Algebraic expressions are critical in effectively solving this problem. They allow us to express relationships between the box's dimensions:

• The height is simply \( h \).
• The width is expressed as \( h + 1 \).
• The length is expressed as \( h + 2 \).

These expressions allow us to write a single equation that represents the box's volume when all dimensions are multiplied together. Algebraic manipulation then involves expanding, simplifying, and solving the equation:\[ h(h^2 + 3h + 2) = 86.625 \].When working with algebraic expressions, it's important to follow mathematical operations properly, expand terms cautiously, and always keep the problem's context in mind. By accurately constructing these expressions, we can approach finding the box dimensions methodically and systematically.