(1 pt) A small resort is situated on an island off a part of the coast of Mexico
ID: 2830937 • Letter: #
Question
(1 pt) A small resort is situated on an island off a part of the coast of Mexico that has a perfectly straight north-south shoreline. The point P on the shoreline that is closest to the island is exactly 3 miles from the island. Ten miles south of P is the closest source of fresh water to the island. A pipeline is to be built from the island to the source of fresh water by laying pipe underwater in a straight line from the island to a point Q on the shoreline between P and the water source, and then laying pipe on land along the shoreline from Q to the source. It costs 2.2 times as much money to lay pipe in the water as it does on land. How far south of P should Q be located in order to minimize the total construction costs? Hint: You can do this problem by assuming that it costs one dollar per mile to lay pipe on land, and 2.2 dollars per mile to lay pipe in the water. You then need to minimize the cost over the interval [0,10] of the possible distances from P to Q.
Explanation / Answer
a = length of pipe on dry land
b = length of pipe in water
To make things easier, we will break vector 'b' up into horizontal and vertical components, x and y.
a + b[y] = 10
b[x] = 3
By Pythagoreans theorem, b[x]^2 + b[y]^2 = b^2.
1a + 2.2b = Cost
a + 2.2 sqrt((b[x])^2 + (b[y])^2) = C
a + 2.2 sqrt((10 - a)^2 + (3)^2) = C
a + 2.2 sqrt( 100 - 20a + a^2 + 9 ) = C
a + 2.2 sqrt( 109 - 20a + a^2 ) = C
Now take the derivative and equate it to 0 to find the optimum value.
C' = 1 + 1.1 (109 - 20a + a^2)^ -1/2 * (2a - 20)
C' = 1 + (1.1 (2a-20) ) / (sqrt (109 - 20a + a^2) )
C' = 1 + (2.2a - 22) / (sqrt (109 - 20a + a^2) )
0 = 1 + (2.2a - 22) / (sqrt (109 - 20a + a^2) )
-1 = (2.2a - 22) / (sqrt (109 - 20a + a^2) )
sqrt (109 - 20a + x^2) = 22 - 2.2a
109 - 20a + a^2 = (22 - 2.2a)^2
109 - 20a + a^2 = 484 - 96.8a + 4.84a^2
0 = 3.84a^2 - 76.8a + 375
Use the quadratic formula:
a = 8.46 miles
10 - 8.46 = 1.54 miles south of point P.
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