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Calculus/Calculus- Minimum problem

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Expert: Alon Mandes - 8/9/2009


Question
Hi,
 I'm having a tough time with this question and would appreciate your advice.
A new cottage is built across the river and 300 m downstream from the nearest telephone relay station. The river is 120 m wide. In order to wire the cottage for phone service, wire will be laid across the river under water and along the edge of the river above ground. The cost to lay wire under water is $15 per m and the cost to lay wire above ground is $10 per m. How much wire should be laid under water to minimize the cost?

Where I'm having difficulty is when I set the derivative to 0 and solve for x. I get x^2=square root(-25950). I can't take the square root of a negative number, so I can't define what the minimum is of the function.
Thank-you.

Answer

drawing
1st of all I attached a drawing to illustrate the problem .
Lets denote z as the distance of underwater to be wired.
Lets denote y as the distance on the ground to be wired.
Pythagoras rule gives us :
(1) y=√[120²-(300-z)√]=√[14400-900+600z-z²]=√[13500+600z-z²] .
Out coast function is :
C(y,z)=15$z+10$y . We substitute eq1 , we get :
C(z)=15z+10√[13500+600z-z²] . All we have to do now is to derive C(z) with respec to z, and
set the derivative equal zero & solve for z :
C'(z)=15 + 2(600-2z)/10√[13500+600z-z²] = 15 + (30000-10z)/10√[13500+600z-z²].
C'(z)=0 --> 15 + (30000-10z)/10√[13500+600z-z²] = 0 -->  -15 = (30000-10z)/10√[13500+600z-z²].
-- > 15=(10z-3000)/10√[13500+600z-z²] --> 150√[13500+600z-z²]=10z-3000 -->
15√[13500+600z-z²]=z-300 . Let's rise both sides of the equation by square operator :
225[13500+600z-z²]=(z-300)²
225[13500+600z-z²]=z²-600z+900
3037500+135000z-225z²=z²-600z+900
-226z²+135600z+3036600=0
226z²-135600z-3036600=0
z1=(135600-125069)/452=23.29
z2=(135600+125069)/452=576.7
To decide which one of the two solutions is min , you have to derive 2nd time & calculate
C''(z1) & C''(z2). Now,
If C''(z1 or z2)<0 then it's maximum
If C''(z1 or z2)<0 then it's maximum

Good luck .

Alon.  

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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