1. Every morming Neo packs his backpack and walks a distance L through a straight path in theforest from his home (A) to the unicorn sanctuary (B). One day he discovers someone has builtan electric fence in the exact middle of his daily path (the dashed, red line in the picture):A•BThe fence has length 2b - it extends for a distance b on each side of the path - and is perpendicularto the path. After a few days, Neo notices that the electric fence is turned on only half the time,but he does not know if the fence is on or off on any given day until he walks up to it and throwshis cat at the fence to test it. If the fence is off, he can just quickly climb over it. Otherwise, hehas to walk around it. He devises a plan: he will walk straight from his home to some point P inthe fence; then, he will walk around it or climb over it depending on whether the fence is on oroff. Which point P should he choose in order to minimize the average length of his trip?

Answered: Which point P should he choose in order…

Algebra for College Students

10th Edition

ISBN:9781285195780

Author:Jerome E. Kaufmann, Karen L. Schwitters

Publisher:Jerome E. Kaufmann, Karen L. Schwitters

Chapter7: Equations And Inequalities In Two Variables

Section7.3: Distance And Slope

Problem 69PS

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Question

1. Every morming Neo packs his backpack and walks a distance L through a straight path in the
forest from his home (A) to the unicorn sanctuary (B). One day he discovers someone has built
an electric fence in the exact middle of his daily path (the dashed, red line in the picture):
A•
B
The fence has length 2b - it extends for a distance b on each side of the path - and is perpendicular
to the path. After a few days, Neo notices that the electric fence is turned on only half the time,
but he does not know if the fence is on or off on any given day until he walks up to it and throws
his cat at the fence to test it. If the fence is off, he can just quickly climb over it. Otherwise, he
has to walk around it. He devises a plan: he will walk straight from his home to some point P in
the fence; then, he will walk around it or climb over it depending on whether the fence is on or
off. Which point P should he choose in order to minimize the average length of his trip?

Expert Solution

Step 1 INTRODUCTION

According to the question, consider a point P at a distance p, above or below, in the fence as shown in figure:

Step 2 SOLUTION

When fence is off, distance travelled = AP+BP

AP+BP= 2 x $\sqrt{{(A O)}^{2} + {(P O)}^{2}}$

= 2 x $\sqrt{\frac{L^{2}}{4} + p^{2}}$ (Because as per diagram, AO= L/2 and PO = p)

when fence is on, distance travelled= AP+PC+CB

AP+PC+CB= $\sqrt{{(A O)}^{2} + {(P O)}^{2}} + b - p + \sqrt{{(C O)}^{2} + {(O B)}^{2}}$

= $\sqrt{{\frac{L}{4}}^{2} + p^{2}} + b - p + \sqrt{{\frac{L}{4}}^{2} + b^{2}}$ (Because as per diagram, PC = b-p, OB= b and CO=L/2)

Probability for fence on or off= 1/2 (equally likely)

Therefore, average distance = $\frac{1}{2} [2 \sqrt{\frac{L^{2}}{4} + p^{2}}] + \frac{1}{2} [\sqrt{\frac{L^{2}}{4} + p^{2}} + b - p + \sqrt{\frac{L^{2}}{4} + b^{2}}]$

D(p) = $\frac{3}{2} \sqrt{\frac{L^{2}}{4} + p^{2}} + \frac{1}{2} \sqrt{\frac{L^{2}}{4} + b^{2}} + \frac{b - p}{2}$

Now, Minimise the equation with respect to p

$\frac{d}{d p} D (p) = 0 = \frac{3}{2} x \frac{1}{2} \frac{2 p}{{(\frac{L^{2}}{4} + p^{2})}^{1 / 2}} + 0 - \frac{1}{2} 3 p = {(\frac{L^{2}}{4} + p^{2})}^{1 / 2} 9 p^{2} = (\frac{L^{2}}{4} + p^{2}) 8 p^{2} = \frac{L^{2}}{4} p = + / - \frac{L}{4 \sqrt{2}}$

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