5. You are starting a race on the beach at point A to reach a buoy at point B in the ocean, as pictured below (bird's eye view). A = (0,0) T(x) = -x + B = (b₁,b₂) You can run along the beach twice as fast as you can swim in the ocean. The aim is to find that the optimal location x to enter the ocean is such that 0 = 30°. Beach Ocean (a) Let v₁ = running speed and v₂ = swimming speed. Show that the total travel time between A and B is (explain your working): 1 VI V2 sin (= x √√(b₁-x)² + b². (b) Calculate T'(x). (c) Show that if v₁ = 2v2, the optimal angle to enter the ocean is 0 = 30°. Hint: Note that T(x) has no maximum (you can take arbitrary long detours) and that (b₁-x) (b₁-x)² + b²
5. You are starting a race on the beach at point A to reach a buoy at point B in the ocean, as pictured below (bird's eye view). A = (0,0) T(x) = -x + B = (b₁,b₂) You can run along the beach twice as fast as you can swim in the ocean. The aim is to find that the optimal location x to enter the ocean is such that 0 = 30°. Beach Ocean (a) Let v₁ = running speed and v₂ = swimming speed. Show that the total travel time between A and B is (explain your working): 1 VI V2 sin (= x √√(b₁-x)² + b². (b) Calculate T'(x). (c) Show that if v₁ = 2v2, the optimal angle to enter the ocean is 0 = 30°. Hint: Note that T(x) has no maximum (you can take arbitrary long detours) and that (b₁-x) (b₁-x)² + b²
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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Hi, could you help me with these maths problems?
Transcribed Image Text:5. You are starting a race on the beach at point A to reach a buoy at point B in the ocean, as
pictured below (bird's eye view).
= (0,0)
A =
X
B = (b₁,b₂)
You can run along the beach twice as fast as you can swim in the ocean. The aim is to find
that the optimal location x to enter the ocean is such that 0 = 30°.
(a) Let v₁ = running speed and v₂ = swimming speed. Show that the total travel time
between A and B is (explain your working):
1
T(x) = -x +
VI
Beach
Ocean
V2
sin 0
√(b₁-x)² + b ²₂.
(b) Calculate T'(x).
(c) Show that if v₁ = 2v2, the optimal angle to enter the ocean is 0 = 30°.
Hint: Note that T(x) has no maximum (you can take arbitrary long detours) and that
(b₁-x)
√√(b₁-x)² + b²
Transcribed Image Text:4. A beam of length 8 m has a deflection y at a distance x from one end given by
y= 03 (x² − 14x² + 36x²).
1
12 x 10³
Determine the maximum deflection, and the value of x to achieve this.
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