Problem 62 - Algebra 2

Question

Write a polynomial function to describe each volume. Then graph your function to solve the problem. The width of a box is \(2 \mathrm{~m}\) less than the length. The height is \(1 \mathrm{~m}\) less than the length. The volume is \(60 \mathrm{~m}^{3}\). Find the length of the box.

Step-by-Step Solution

Answer

The length of the box is approximately 5 metres.

Step 1: Define the Problem

Translate the word problem into an equation. We know that the volume \( V \) of a rectangular box is given by \( V = lwh \) where \( l \) is the length, \( w \) is the width, and \( h \) is the height.\nSince the width is 2m less than the length, \( w \) can be expressed as \( l-2 \). \nThe height is 1m less than the length, so \( h \) can be expressed as \( l-1 \). \nSubstitute these values into the equation for volume: \( V = l(l-2)(l-1) \)

Step 2: Apply Given Volume

We know that the volume of the box is 60 cubic meters, so we can substitute \( V = 60 \) into the equation. We then end up with the equation: \( 60 = l(l-2)(l-1) \)

Step 3: Simplify the Equation

Expand the brackets to simplify the equation: \( 60 = l^3 - 3l^2 + 2l \)

Step 4: Solve the Equation

As this is a cubic equation, it can be complex to solve. However, given that lengths are physical quantities and can't be negative, we're looking for the positive real root of this equation. After solving for \( l \) using methods like factorisation or Newton-Raphson method, you determine that the length of the box is approximately 5 metres.

Step 5: Verification

By substituting the calculated length back into your original equation for the volume, ensure that it matches up with the given volume of 60 cubic meters.

Key Concepts

Cubic Equations

Cubic equations are polynomials where the highest degree is three, which means the term with the largest exponent is raised to the third power. These equations generally take the form \( ax^3 + bx^2 + cx + d = 0 \). In the context of the given problem, our cubic equation emerges when we decipher the volume of a box in terms of its length. By expressing the width and height as functions of the length \( l \), we end up with the cubic equation \( l^3 - 3l^2 + 2l = 60 \). To find the appropriate length of the box that satisfies the volume, we need to solve this cubic equation. This could potentially involve different solution methods like factoring or numerical methods such as the Newton-Raphson method, given that it can be challenging to find exact roots. In this case, we focus on finding the positive real root since we're dealing with physical dimensions.

Volume of a Box

The volume of a rectangular box is calculated using the formula \( V = lwh \), where \( l \) stands for length, \( w \) for width, and \( h \) for height. In the problem, we're given additional relationships for \( w \) and \( h \). The width is \( 2 \) meters less than the length \( l \), so it's represented as \( l-2 \). Similarly, the height is \( 1 \) meter less than the length, represented as \( l-1 \). By inserting these into the volume formula, the equation becomes dependent on one variable: \( V = l(l-2)(l-1) \). Here, the variable \( l \) controls all dimensions of the box. This equation reveals how each dimension impacts the overall volume, providing a clear algebraic approach to solving the problem of determining the size of each dimension when the volume is fixed.

Graphing Polynomials

Graphing polynomials can provide a visual perspective on the properties of the polynomial function, especially when dealing with higher degrees like cubic equations. By graphing the function \( f(l) = l^3 - 3l^2 + 2l - 60 \), you can visually inspect for roots where the graph crosses the x-axis, indicating the values that satisfy the equation \( f(l) = 0 \). The positive root found by graphing helps identify the length of the box.
Graphs of cubic functions are typically characterized by having an "S" shape, with up to three roots depending on the specific function. By examining the graph, students can identify intercepts, optimal points, and regions where the function is increasing or decreasing, thus gaining insight into the nature and solution of the polynomial equation.

Solving Equations

Solving equations, particularly cubic ones, involves determining the values that satisfy the given algebraic expression. Here, we're specifically interested in solving \( l^3 - 3l^2 + 2l - 60 = 0 \) to find the length of the box that leads to a volume of 60 cubic meters.

• One method could be factorization, which, however, is often difficult, especially when no obvious roots are present.
• Another approach is using algorithms such as the Newton-Raphson method, which iteratively approximates roots. This can be effective for finding real and positive roots quickly.

It's crucial to verify your obtained solution by substituting back into the original volume expression. This not only confirms the solution's correctness but also reinforces understanding by checking that the results align with the physical constraints and given data of the problem.