Problem 61 - Precalculus : Building Concepts and Connections

Question

You will use polynomials to study real-world problems. Geometry The length of a rectangular box is 10 inches more than the height, and its width is 5 inches more than the height. Find the dimensions of the box if the volume is 168 cubic inches.

Step-by-Step Solution

Answer

Therefore, the physical dimensions of the box (H x W x L) are approximately 3 x 8 x 13 inches.

Step 1: Set up the equations

Using the problem's data, you can write the equations as follows: L = H + 10, W = H + 5. This denotes that both the length and width of the rectangular box are functions of the height.

Step 2: Substitute the values of L and W into the volume equation

You can set the volume of the box V equal to the product of L, W and H which gives you V = L * W * H. Substitute the values from the previous step into this equation to get: 168 = (H+10)*(H+5)*H.

Step 3: Form a cubic equation

This equation expands to form a cubic equation: \(168 = H^3 + 15H^2 +50H\).

Step 4: Solve the cubic equation

Rearrange the equation to make it equals to zero: \(H^3 + 15H^2 + 50H - 168 = 0\). Now solve this cubic equation for height

Step 5: Finding the roots

A cubic equation can have maximum of 3 real roots. Here, for the values to make sense in the physical world, the height of the box should be greater than 0. Solve the cubic equation using any numerical method. One obtainable real root which makes sense in the physical context of the problem is around H=3.

Key Concepts

Cubic equations

Cubic equations are polynomial equations of degree three. They usually take the form \(ax^3 + bx^2 + cx + d = 0\), where \(a, b, c,\) and \(d\) are constants, and \(a eq 0\). In this exercise, the cubic equation was derived from substituting geometric expressions into the volume equation of a rectangular box. This allowed for the formation of the equation \(H^3 + 15H^2 + 50H - 168 = 0\).
Solving cubic equations can sometimes be challenging, but it’s essential to understand that they have up to three real roots or solutions. These roots may represent real values when applied to problems such as geometry, where you are solving for physical dimensions.
Numerical methods like the Rational Root Theorem, synthetic division, or graphing calculators can help find these roots, especially when factoring is not straightforward. In our exercise, we found the root \(H = 3\), which is logical when applying the context of the problem.

Volume calculations

Volume calculations are crucial in understanding how to determine the capacity or space that an object occupies. For a rectangular box, the volume \(V\) is calculated using the formula \(V = L \times W \times H\), where \(L\), \(W\), and \(H\) represent the length, width, and height respectively.
In the problem scenario, the box’s volume was given as 168 cubic inches, so the calculations focused on expressing \(L\) and \(W\) in terms of \(H\). By substituting these expressions into the volume formula, a cubic equation in terms of \(H\) was formed.
It's important to carefully substitute and simplify such expressions to avoid errors in calculation. Understanding how to manipulate and solve these equations allows us to find accurate dimensions of the objects we're studying, just like finding the dimensions of the box in this exercise.

Real-world applications

The techniques used in this exercise have practical applications in real-world scenarios. Calculating volumes and solving equations related to physical measurements are common in various fields, such as engineering, architecture, and manufacturing.
Understanding and solving cubic equations can help in designing objects that meet specific volume requirements or determining the dimensions of existing structures. In construction, for instance, knowing how to calculate the size needed for a shipping container or storage room based on a set volume is invaluable.

• Applications extend to designing furniture, packaging products, and developing structures with particular capacity constraints.
• Proficiency in these calculations ensures accurate planning and efficient material use.
• Real-world problems often require creative thinking to form appropriate equations that reflect the given conditions accurately.

Mastering these concepts ensures you can tackle both theoretical and practical challenges effectively.