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Chapter 1: Problem 68

Perform the indicated operations. In designing a building, it was determined that the forces acting on an I beam would deflect the beam an amount (in \(\mathrm{cm}\) ), given by \(\frac{x\left(1000-20 x^{2}+x^{3}\right)}{1850},\) where \(x\) is the distance (in \(m\) ) from one end of the beam. Find the deflection for \(x=6.85 \mathrm{m}\). (The 1000 and 20 are exact.)

Short Answer

Expert verified
The deflection is approximately 1.42 cm.

Step by step solution

01

Substitute the given value into the formula

We need to find the deflection for \(x = 6.85 \, \mathrm{m}\). Substitute \(x = 6.85\) into the formula \(\frac{x\left(1000-20x^{2}+x^{3}\right)}{1850}\). This gives us \(\frac{6.85\left(1000 - 20(6.85)^{2} + (6.85)^{3}\right)}{1850}\).
02

Calculate the powers of x

Calculate \( (6.85)^2 \) and \( (6.85)^3 \). First, find \( (6.85)^2 = 46.9225 \) and \( (6.85)^3 = 321.489125 \).
03

Simplify the expression inside the parentheses

Substitute the calculated values into the expression: \(1000 - 20(46.9225) + 321.489125\). Calculate \(- 20(46.9225) = -938.45\). Then simplify: \(1000 - 938.45 + 321.489125 = 383.039125\).
04

Multiply by x

Multiply \(x = 6.85\) by the result from Step 3: \(6.85 \times 383.039125 = 2623.01760625\).
05

Divide by 1850

Finally, divide the result from Step 4 by 1850 to get the deflection: \(\frac{2623.01760625}{1850} \approx 1.417303\, \mathrm{cm}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Calculus
Calculus is an essential branch of mathematics that helps us understand how things change. In the context of an I-beam's deflection in a building, calculus allows us to calculate how far the beam bends under certain forces. This problem involves a polynomial function to determine deflection. Here, derivatives, integrals, and polynomials become fundamental tools. Understanding calculus facilitates solving real-world problems involving dynamic change, such as material deflection over distance.
  • **Derivatives**: They represent the rate of change of a function. In beam design, derivatives help find the slope and predict how quickly deflection will change concerning distance.
  • **Integrals**: These are important for calculating aspects like total area under a curve, which in physics translates to total force on a beam.
By calculating the deflection at specific points, using steps like squaring and cubing values for 'x', calculus provides exact mathematical solutions to engineering challenges.
Physics
Physics plays a pivotal role in understanding the behavior of materials and structures. When a force is applied to an I-beam, physicists use principles like elasticity and deflection to predict movement. This problem brings in the physical aspect by using a mathematical model to indicate how a force affects a beam.
  • **Elasticity**: This concept explains how materials return to their original form after deformation. Beam elasticity impacts the calculations of deflection, as it determines how far a beam can bend before returning to its initial state.
  • **Force and Pressure**: In beams, distributed force dramatically impacts how much they bend. Models relying on physics allow us to quantify these effects precisely, assessing safety margins in engineering.
In the context of work like this, advanced physics concepts are simplified into equations, allowing engineers to forecast and mitigate structural challenges.
Engineering
Engineering melds physics and calculus to innovate and design. In structural engineering, understanding how elements like beams respond to loads and forces is essential for safe construction. Engineers must ensure beams like the I-beam in this problem deflect within allowable limits.
  • **Structural Safety**: Engineering principles ensure that the beam's deflection doesn't exceed safe limits, affecting a building's structural integrity.
  • **Precision Design**: Calculating specific deflection requires precise measurements and consideration of all variables, ensuring designs meet rigorous standards.
  • **Material Selection**: Different materials have varying deflection rates, which affects design choices engineers make to optimize performance and cost efficiency.
By using calculations in structural engineering processes, engineers design buildings that are safe, efficient, and stand the test of time, meeting both practical and regulatory standards.
Problem Solving
Problem-solving in calculus-based scenarios involves identifying the best approaches to break down complex tasks into manageable steps. In finding deflection, each step builds towards the final solution. Understanding the sequence and rationale behind each operation enhances learning and application.
  • **Step-by-Step Approach**: Decomposing the problem by substituting numerical values and simplifying expressions helps clarify the path to a solution.
  • **Logical Reasoning**: Following a clear logical sequence ensures every calculation aligns with the prior steps, minimizing errors.
  • **Accuracy and Verification**: Double-checking calculations ensures results are reliable, especially for safety-critical applications in engineering.
Approaches to problem-solving blend systemic thinking with mathematical skills, ensuring that solutions are comprehensive, accurate, and applicable across real-life scenarios.

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