Logo

AskSia

Plus

13 If a surveyor measures differences in elevation when making plans for a high...
Aug 5, 2024
Solution by Steps
step 1
To show that the correction CC is C=Rsec⁡(LR)−RC = R \sec \left(\frac{L}{R}\right) - R, we start with the given formula
step 2
Using the definition of secant, sec⁡(x)=1cos⁡(x)\sec(x) = \frac{1}{\cos(x)}, we can rewrite the formula as C=R(1cos⁡(LR))−RC = R \left(\frac{1}{\cos\left(\frac{L}{R}\right)}\right) - R
step 3
Simplifying, we get C=Rsec⁡(LR)−RC = R \sec\left(\frac{L}{R}\right) - R
Answer
C=Rsec⁡(LR)−RC = R \sec\left(\frac{L}{R}\right) - R
Key Concept
Secant function in trigonometry
Explanation
The secant function is the reciprocal of the cosine function, which helps in deriving the correction formula for the curvature of the Earth.
Solution by Steps
step 1
To show the Maclaurin series for CC, we start with the Taylor series expansion of sec⁡(x)\sec(x) around x=0x = 0
step 2
The Taylor series for sec⁡(x)\sec(x) is 1+x22+5x424+O(x6)1 + \frac{x^2}{2} + \frac{5x^4}{24} + O(x^6)
step 3
Substituting x=LRx = \frac{L}{R}, we get sec⁡(LR)=1+L22R2+5L424R4+O(L6R6)\sec\left(\frac{L}{R}\right) = 1 + \frac{L^2}{2R^2} + \frac{5L^4}{24R^4} + O\left(\frac{L^6}{R^6}\right)
step 4
Therefore, C=R(1+L22R2+5L424R4+O(L6R6))−RC = R \left(1 + \frac{L^2}{2R^2} + \frac{5L^4}{24R^4} + O\left(\frac{L^6}{R^6}\right)\right) - R
step 5
Simplifying, we get C≈L22R+5L424R3C \approx \frac{L^2}{2R} + \frac{5L^4}{24R^3}
Answer
C≈L22R+5L424R3C \approx \frac{L^2}{2R} + \frac{5L^4}{24R^3}
Key Concept
Maclaurin series expansion
Explanation
The Maclaurin series is a special case of the Taylor series centered at zero, used to approximate functions.
Solution by Steps
step 1
Using the formula C=Rsec⁡(LR)−RC = R \sec\left(\frac{L}{R}\right) - R with R=6370R = 6370 km and L=100L = 100 km, we calculate CC
step 2
Substituting the values, C=6370sec⁡(1006370)−6370C = 6370 \sec\left(\frac{100}{6370}\right) - 6370
step 3
Using a calculator, sec⁡(1006370)≈1.000785\sec\left(\frac{100}{6370}\right) \approx 1.000785
step 4
Therefore, C≈6370×1.000785−6370≈5.000785C \approx 6370 \times 1.000785 - 6370 \approx 5.000785
step 5
Using the Maclaurin series C≈L22R+5L424R3C \approx \frac{L^2}{2R} + \frac{5L^4}{24R^3}, we substitute L=100L = 100 km and R=6370R = 6370 km
step 6
Calculating, C≈10022×6370+5×100424×63703≈0.785C \approx \frac{100^2}{2 \times 6370} + \frac{5 \times 100^4}{24 \times 6370^3} \approx 0.785
Answer
Using the exact formula: C≈5.000785C \approx 5.000785 km
Using the Maclaurin series: C≈0.785C \approx 0.785 km
Key Concept
Comparison of exact and approximate corrections
Explanation
The exact formula provides a more precise correction, while the Maclaurin series offers a simpler approximation that is less accurate for larger values of LL.
© 2023 AskSia.AI all rights reserved