Problem 56
Question
There is a relationship between the length of a suspension bridge cable that is secured between two vertical supports and the amount of sag of the cable. If we represent the length of the cable by \(c,\) the horizontal distance between the vertical supports by \(d,\) and the amount of sag by \(s,\) the equation is \(c=d+\frac{8 s^{2}}{3 d}-\frac{32 s^{4}}{5 d^{3}} .\) If the horizontal distance between the two vertical supports is 190 feet and the amount of sag in a cable that is suspended between the two supports is 20 feet, what is the length of the cable?
Step-by-Step Solution
Verified Answer
Answer: The approximate length of the suspension bridge cable is 195.465 feet.
1Step 1: Write down the information given
We know the following:
- Horizontal distance between supports (\(d\)) = 190 feet
- Amount of sag (\(s\)) = 20 feet
- The equation relating cable length (\(c\)), horizontal distance (\(d\)), and sag (\(s\)):
\(c=d+\frac{8 s^{2}}{3 d}-\frac{32 s^{4}}{5 d^{3}}\)
2Step 2: Substitute the given values into the equation
Place the known values of \(d\) and \(s\) in the equation:
\(c=190+\frac{8(20)^{2}}{3(190)}-\frac{32(20)^{4}}{5(190)^{3}}\)
3Step 3: Calculate the fractions
Simplify the equation by computing the fractions:
\(c=190+\frac{8(400)}{(3)(190)}-\frac{32(160000)}{5(6859000)}\)
\(c=190+\frac{3200}{570}-\frac{5120000}{34295000}\)
4Step 4: Solve for the cable length (\(c\))
Combine the fractions and solve for \(c\):
\(c=190+5.614-\frac{5120000}{34295000}\)
\(c=195.614-\frac{5120000}{34295000}\)
Now we can convert the remaining fraction into decimal and add it to the equation to get the length of the cable:
\(c = 195.614 - 0.149\)
\(c \approx 195.465\)
So, the length of the suspension bridge cable is approximately 195.465 feet.
Key Concepts
Algebraic Problem-SolvingEquation Solving in AlgebraMathematical Modeling
Algebraic Problem-Solving
Algebraic problem-solving is a critical skill in mathematics that involves finding unknowns within the confines of given relationships. For example, when solving the length of a suspension bridge cable, it's essential to understand the algebraic relationship that connects cable length (\(c\)), horizontal distance (\(d\)), and sag (\(s\)).
Students start by identifying known values and substitute them into an algebraic expression. In our case, the textbook provides an equation that encapsulates the physics of a suspension cable. Through careful substitution and accurate arithmetic, one can resolve the problem efficiently. An important point is ensuring units are consistent; here, feet is standard, thus making calculations straightforward. This method of breaking down the original complex problem into simpler arithmetic operations is a cornerstone of algebraic problem-solving.
When students convert the complex fractions into decimals during calculations, it's particularly helpful for visualizing the answer more clearly, which demonstrates another essential aspect of problem-solving: simplification for the sake of clarity and ease.
Students start by identifying known values and substitute them into an algebraic expression. In our case, the textbook provides an equation that encapsulates the physics of a suspension cable. Through careful substitution and accurate arithmetic, one can resolve the problem efficiently. An important point is ensuring units are consistent; here, feet is standard, thus making calculations straightforward. This method of breaking down the original complex problem into simpler arithmetic operations is a cornerstone of algebraic problem-solving.
When students convert the complex fractions into decimals during calculations, it's particularly helpful for visualizing the answer more clearly, which demonstrates another essential aspect of problem-solving: simplification for the sake of clarity and ease.
Equation Solving in Algebra
Equation solving is at the heart of algebra. It often entails manipulating expressions to isolate the unknown variable. For instance, in our suspension bridge cable length problem, the variable \(c\) that represents the cable length is isolated through a series of steps that simplify and eliminate fractions.
One common technique is to perform operations that ultimately cancel out terms to simplify the equation. This may involve operations like addition, subtraction, multiplication, division, or a combination of these. As seen in the provided solution, fractions are combined and simplified, eventually converting one into a decimal to facilitate the final subtraction, which exemplifies these algebraic principles in action.
One common technique is to perform operations that ultimately cancel out terms to simplify the equation. This may involve operations like addition, subtraction, multiplication, division, or a combination of these. As seen in the provided solution, fractions are combined and simplified, eventually converting one into a decimal to facilitate the final subtraction, which exemplifies these algebraic principles in action.
Key Arithmetic Steps
As part of the equation solving process, it's vital to perform the arithmetic with precision. This includes calculating squared values and understanding the impact of adding and subtracting terms to move closer to the final solution.Mathematical Modeling
Mathematical modeling uses algebra to represent real-world situations, like the construction of a suspension bridge. The equation given for the cable length \(c = d + \frac{8s^2}{3d} - \frac{32s^4}{5d^3}\) is derived from physics principles specific to suspended structures.
In the context of educational tools, mathematical models enable students to see the relevance of abstract algebra in everyday engineering challenges. Understanding how to derive and manipulate such models is essential, as it improves cognitive ability to translate real-world complexities into solvable equations.
The model's accuracy is also a point for consideration. While it accurately describes the cable behavior within certain constraints, factors like material properties or environmental conditions might be simplified or omitted for the educational exercise. Highlighting this aspect can help students appreciate the model's context and limitations.
In the context of educational tools, mathematical models enable students to see the relevance of abstract algebra in everyday engineering challenges. Understanding how to derive and manipulate such models is essential, as it improves cognitive ability to translate real-world complexities into solvable equations.
The model's accuracy is also a point for consideration. While it accurately describes the cable behavior within certain constraints, factors like material properties or environmental conditions might be simplified or omitted for the educational exercise. Highlighting this aspect can help students appreciate the model's context and limitations.
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