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The position of a drone at time t≥ 0 (in minutes) is given by the parametric function. 4nt x = a(t) cos(3nt) + b(t)cos | 4πt ya(t) sin(3πt) + b(t)sin 3

The position of a drone at time t≥ 0 (in minutes) is given by the parametric function. 4nt x = a(t) cos(3nt) + b(t)cos | 4πt ya(t) sin(3πt) + b(t)sin 3

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question The position of a drone at time t≥ 0 (in minutes) is given by the parametric function. x = a(t) cos(3nt) + b(t)cos 4πt y = a(t) sin(3πt) + b(t)sin (4) where a(t) and b(t) are functions of time. Both x and y are measured in metres. a) Show the squared distance from the origin, i.e. d² = x² + y²,can be written d² = (a(t))² + (b(t))² + 2a(t)b(t)cos ( b) If the functions a(t) and b(t) 5πt 5πt a(t) = 4+2cos 3 5πt b(t) = 1 -4cos (5) now show the squared distance from the origin, i.e. d² = x² + y², can be rewritten as d² = -16 cos³ -8 cos² 5nt '5πt' +16cos 5πt +17 c) Using (b) find algebraically all times, t≥0 (in exact form i.e. in terms of fractions) when the distance d is 3 mi.e. d = 3 or d² = 9. Hint. periodic solutions can be represented in the form t = a + nT where a is one solution, T is the period, and n is an integer n = 0,1 ....
Transcribed Image Text:Question The position of a drone at time t≥ 0 (in minutes) is given by the parametric function. x = a(t) cos(3nt) + b(t)cos 4πt y = a(t) sin(3πt) + b(t)sin (4) where a(t) and b(t) are functions of time. Both x and y are measured in metres. a) Show the squared distance from the origin, i.e. d² = x² + y²,can be written d² = (a(t))² + (b(t))² + 2a(t)b(t)cos ( b) If the functions a(t) and b(t) 5πt 5πt a(t) = 4+2cos 3 5πt b(t) = 1 -4cos (5) now show the squared distance from the origin, i.e. d² = x² + y², can be rewritten as d² = -16 cos³ -8 cos² 5nt '5πt' +16cos 5πt +17 c) Using (b) find algebraically all times, t≥0 (in exact form i.e. in terms of fractions) when the distance d is 3 mi.e. d = 3 or d² = 9. Hint. periodic solutions can be represented in the form t = a + nT where a is one solution, T is the period, and n is an integer n = 0,1 ....
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d) What are the first four times, after t = 0, that the distance d of the drone from the origin is 3m? Give your answers exactly (as a fraction) and approximately as a decimal.
Transcribed Image Text:d) What are the first four times, after t = 0, that the distance d of the drone from the origin is 3m? Give your answers exactly (as a fraction) and approximately as a decimal.
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