Problem 40 - Basic Technical Mathematics with Calculus

Question

Solve the given problems. Use a calculator to solve if necessary. A rectangular safe is to be made of steel of uniform thickness, including the door. The inside dimensions are \(1.20 \mathrm{m}, 1.20 \mathrm{m},\) and 2.00 \(\mathrm{m}\). If the volume of steel is \(1.25 \mathrm{m}^{3}\), find its thickness.

Step-by-Step Solution

Answer

The steel thickness is approximately 0.05 meters.

Step 1: Determine the outer dimensions

To find the thickness, we need to know the outer dimensions of the safe. Let the thickness be \( t \). Then, the outer dimensions will be \( (1.20 + 2t) \times (1.20 + 2t) \times (2.00 + 2t) \).

Step 2: Calculate the outer volume

The outer volume of the safe, including the steel, can be calculated using the outer dimensions: \[V_{outer} = (1.20 + 2t)(1.20 + 2t)(2.00 + 2t). \]

Step 3: Calculate the inner volume

The inner volume is given by the inner dimensions of the safe:\[V_{inner} = 1.20 \times 1.20 \times 2.00 = 2.88 \, \text{m}^3.\]

Step 4: Use the volume of steel given

The volume of steel is the difference between the outer volume and the inner volume, which is given as 1.25 \( \text{m}^3 \) :\[V_{steel} = V_{outer} - V_{inner} = 1.25 \, \text{m}^3.\]

Step 5: Set up the equation for thickness

Substitute into the equation from Step 4:\[(1.20 + 2t)^2(2.00 + 2t) = 2.88 + 1.25.\]Simplify this to solve for \( t \).

Step 6: Solve for thickness

First, calculate the right-hand side:\[2.88 + 1.25 = 4.13.\]Next, solve the equation \((1.20 + 2t)^2(2.00 + 2t) = 4.13\) by expanding and approximating to solve for \( t \). Use a calculator for precise numerical solution.

Key Concepts

Volume

Volume is a measure of the amount of space an object occupies, and in geometry, it is crucial for understanding three-dimensional shapes. For any 3D object, like the rectangular safe in the exercise, volume is calculated by multiplying its length, width, and height.
To solve problems involving volume:

• Identify whether you need to find the volume of the entire shape or a specific part like the material or the hollow space inside.
• Use the formula for volume appropriate to the shape, and don't forget to incorporate any thickness or external layers if needed.
• Be cautious with units, ensuring all measurements are in the same unit before performing calculations.

Using these steps efficiently helps avoid errors and ensures that the calculation is accurate.

Rectangular Prism

A rectangular prism is a three-dimensional geometric figure with six faces, all of which are rectangles.
Key characteristics include:

• Opposite faces are always equal in area.
• The volume of a rectangular prism is found by multiplying its length, width, and height: \[ V = l \times w \times h. \]
• The surface area can be found by calculating the sum of the areas of all six rectangles.

This shape is commonly seen in everyday objects, such as boxes or safes, as in the exercise. Understanding this basic shape is essential for navigating to more complex problems involving three-dimensional geometric modeling.

Calculus

Calculus is a branch of mathematics focused on change and motion, through concepts of derivatives and integrals. In the context of this exercise, although calculus per se isn't directly applied, understanding calculus concepts can be helpful in solving complex volumetric and geometric problems.
Some basics to keep in mind:

• Differentiation could be useful for finding rate of change, which is valuable for varying dimensions in modeling.
• Integration helps in calculating areas under curves or the accumulation of quantities.

While this particular exercise primarily utilizes algebra, calculus might be necessary for extended problems where dimensions change dynamically or when exploring optimization scenarios.

Mathematical Modeling

Mathematical modeling involves creating a mathematical representation of a real-world scenario, like the steel thickness problem we're discussing. It allows you to analyze and predict behaviors of complex systems.
Steps in mathematical modeling:

• Define the problem and its constraints. In this case, the inner volume known and the volume of steel provided.
• Construct a mathematical equation which represents the situation, such as using the outer volume formula for the safe.
• Simplify the model or equations to make solving easier. This involves using algebra to solve for unknowns like the thickness in the exercise.

By following a systematic approach to modeling, challenging problems become manageable and solvable.