Math Problem Statement
Solution
The question concerns modeling and calculating the properties of the cable curve of the Golden Gate Bridge using a quadratic function. Let's break this into parts:
Given:
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Function form: (quadratic polynomial for the cable's shape).
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Key Points:
- Anchor points at and (cable starts and ends 220 m above the water).
- The lowest point of the cable is at , where .
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Cable length formula:
The arc length of from to is given by:
(a) Find the equation for :
The equation has three unknowns (). We use the given points to form equations:
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At :
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At : Simplifies to: 1638400a + 1280b = 0 \tag{1}
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At : Simplifies to: 409600a + 640b = -140 \tag{2}
Solve for and :
From :
b = -1280a \tag{3}
Substitute into :
Substitute into :
Thus:
(b) Calculate the cable length:
The arc length is:
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Derivative :
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Simplify :
This integral can now be evaluated numerically to find the cable length.
Would you like a detailed calculation of the integral or any clarification?
Related Questions:
- How does a quadratic polynomial model a parabolic cable shape accurately?
- What assumptions are made in calculating arc length of a cable?
- How do you derive and compute arc length step by step?
- How does the placement of anchor points affect the shape of ?
- What is the impact of numerical errors in approximating cable length?
Tip: Always double-check units when working with physical models to ensure consistency across calculations!
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Math Problem Analysis
Mathematical Concepts
Quadratic Functions
Arc Length
Differentiation
Calculus
Formulas
f(x) = ax^2 + bx + c
Arc Length = ∫(x1 to x2) √(1 + (f'(x))^2) dx
f'(x) = 2 * (7/20480)x - 7/16
Theorems
Arc Length Theorem
Differentiation of Quadratic Functions
Suitable Grade Level
Grades 11-12
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