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Differential Equations/application of differentiation

Question

installation of teleph
determine the least expensive route for telephone cable which connects A and B as in figure.If it costs RM5000 per km to lat the cable on land and RM8000 per km to lay the cable across the river,find the least expensive route.

We need to determine an equation for the total cost.

The length across the river is a hypotenuse of a triangle with length x and 2.
It then can be said to have length √(x²+2²).  For this, the cost is then RM80000√(x²+4).

The length along the bank is (8-x).  The cost is then RM5000(8-x).

This makes the total cost equation be C(x) = RM80000√(x²+4) + RM5000(8-x).
Just to simplify this and make the numbers easier to work with, we can factor out RM1000.
This C(x) = [RM1000][8√(x²+4) + 5(8-x)], so C(x) = K[8√(x²+4) + 5(8-x)] where K = RM1000.

From here, determine C'(x), set it equal to 0, and solve for x.

Now it can be seen that C'(x) = K([8(1/2)/√(x²+4)][2x] - 5).
Simplifying that and setting it to 0 gives C'(x) = 8x/√(x²+4) - 5 = 0
That means (after adjusting the equation) that 8²x² = 5²(x²+4), or 64x² = 25x² + 100.
Subtracting the 25x² from both sides gives 39x² = 100.
Dividing by 39 and taking the square-root of both sides gives x = √(100/39) = 1.601281538.

To make sure that this is correct, put in x+0.1 and x-0.1.
At that value, I get f(x) = 57.49.
At f(x+0.01) and f(x-0.01), I get 57.50.

Remembering to adjust for out value of K, I get C(x) = 57,490.

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Differential Equations

Volunteer

Scott A Wilson

Expertise

I am capable of solving most ordinary differential equations and a few not so ordinary, but those are few and far between.

Experience

I have taken Differential Equations at OSU. Occasionally I have assisted people with questions on the subject.

Publications
I wrote a thesis on the study of shock waves and rarefaction fans. That occurs in nature when mathematics fails to apply. See, mathematics says that when shock waves appear, there should be two solutions whereas nature knows there is only one way to be in that area. When rarefaction fans occur, the mathematics says there should be no solution, but despite this, nature still moves ahead with its own plan.

Education/Credentials
I received an MS degree at OSU in Mathemematics and a BS degree at OSU in Mathematical Science.

Awards and Honors
I graduated from OSU with a Bachelor of Science degree in mathematics with honors. Two years later, I graduated from OSU with a Master of Science degree in mathematics with honors. Between these two degrees, I have more hours in graduate courses that required to get them.

Past/Present Clients
I answered many questions at OSU, down in south seattle at a college, at a church in Corvallis, OR, as Safeway in Washington, and many other areas. I have answered over 8,500 right here, but only a little over 190 of them have been in differential equations.