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Calculus/max- min optimization
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Question
a lighthouse is 8 miles off a straight coast and a town is located 18 miles down the seacoast. supplies are to be delivered at a minimum time. if supplies can be moved at the rate of 7 miles/hour on water and 25 miles/hour on land, how far from the town should a dock be constructed for shipment of supplies?
Let's set y as the distance from the dock to the town. The amount of
time t for the supplies will be t=t1+t2. Where t1 is the time in
water & te is the time on land. Suppose that the supplies travels x1
miles in water & x2 miles on land , thus :
1. t1=x1/7
2. t2=x2/25
We also know that x2=sqrt[8^2+(18-y)^2] according to Pythagoras.
Hence :
t=sqrt[8^2+(18-y)^2]/7+y/25 .
t'=-(18-y) / { 7*sqrt[8^2+(18-y)^2] } + 1/25.
t'=0 -> (18-y) / { 7*sqrt[8^2+(18-y)^2] } = 1/25 ->
[25/7]^2 * (18-y)^2 = 8^2+(18-y)^2. To solve this quickly, let's set
z=18-y . This gives us :
[25/7]^2 * z^2 = 8^2+z^2 -> z^2=8^2/{[25^2/7^2] -1} ->
y=15.6666 miles.
Alon.
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Answers by Expert:
Alon Mandes
Expertise
Kind of questions I can answer : Limits, Derivatives, Integration, Implicit functions, continuousity, differentiation ,Extremum problems, Lagrange multipliers, Gradients, Surface integrals, Multi variables functions ,Multi variables Integrals,Complex variables ,Complex functions, Curves, Trajectory integrals & Vector analyse,Divergence,Rotor & word problems. Kind of question I can't answer : Economics,Combinatorics,infinite series & convergence ,Statistics & Probabilities .
Experience
1. I'm a team member of mathnerds (math site for answering questions)
2. I'm a team member in the Student's Union of the Technion, helping
students who have problems in mathematics.
3. 2 years of experience as a math teacher in college.
4. I give free homework help for high school students in
Mathematics & Physics.
5. I teach part time in collage the subjects : "Digital Signal Processing" , "Random Signals & Noise" ,
"Complex Functions".
Organizations
Hi-Tech company : GSM4VOIP ; job possition : Algorythm developer.
Education/Credentials
M.A in Mathematics & Bs.c in Electronics.
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