anonymous
  • anonymous
Two towns, A and B, are 13.0 miles apart and located 8.0 and 3.0 miles east, respectively, of a long, straight highway. A construction company has a contract to build a road from town A to the highway and then to town B. Determine the length (to the nearest tenth of a mile) of the shortest road that meets these requirements.
Calculus1
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
@ganeshie8 can you draw it picture,so that i can solve?
ganeshie8
  • ganeshie8
|dw:1436781239171:dw|
ganeshie8
  • ganeshie8
|dw:1436782203008:dw|

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ganeshie8
  • ganeshie8
I think we need to find the minimum length of that road
anonymous
  • anonymous
did they told us that the road should meet at the same point to highway?
anonymous
  • anonymous
@ManhattanProject do you have any idea about answer?
dumbcow
  • dumbcow
|dw:1437272681909:dw| now express distance of 2 roads as function of x \[f(x) = \sqrt{x^2 +8^2} + \sqrt{(12-x)^2 +3^2}\] minimize function by taking derivative and setting equal to 0 (use chain rule) \[f'(x) = \frac{x}{\sqrt{x^2+64}} - \frac{12-x}{\sqrt{(12-x)^2 + 9}} = 0\] Solve for x first square both sides \[\frac{x^2}{x^2+64} = \frac{(12-x)^2}{(12-x)^2 +9}\] \[x^2 (x^2 -24x +153) = (x^2 +64)(x^2 -24x+144)\] \[57x^2 -(64)(24)x +(64)(144) = 0\] \[19x^2 - 512x+3072 = 0\] At this point just plug it into quadratic formula (remember x < 12) \[x = 9.02\] Now plug this value into f(x) to get length of shortest road

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