( Hint : The length of CS is StartRoot 16 plus x squared EndRoot16+x2.) A.) S is
ID: 2882603 • Letter: #
Question
(Hint: The length of CS is StartRoot 16 plus x squared EndRoot16+x2.)
A.) S is _______ miles from A. (Round to 2 decimal places)
A power line is to be constructed from a power station at point A to an island at point C, which is 44 mi directly out in the water from a point B on the shore. Point B is 88 mi downshore from the power station at A. It costs $54005400 per mile to lay the power line under water and $30003000 per mile to lay the line under ground. At what point S downshore from A should the line come to the shore in order to minimize cost? Note that S could very well be B or A.(Hint: The length of CS is StartRoot 16 plus x squared EndRoot16+x2.)
A.) S is _______ miles from A. (Round to 2 decimal places)
Explanation / Answer
Using the diagram in the text, let x be the distance to the point at which the power line will reach land. Then the
distance under water is 44 + x2, giving a cost of 540044 + x2 for this part of the line.
(Remember: If the sides of a right triangle have lengths a and b and the hypotenuse has length c, the Pythagorean
theorem says that c2 = a2 + b2 .)
The remainder of the line is on land, so the remaining length costs 3000(88 x).
Cost function: C(x) = 5400(44 + x2 )+ 3000(88 x)
C'(x) = 5400*x / (44 + x 2 ) - 3000
put C'(x) = 0
5400*x / (44 + x 2 ) = 3000
squaring both sides
(54002*x2)=30002 (44 + x 2)
x = 4.4320 mi
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