✔ 最佳答案
Let the path of the cable be along the coast from Q towards P for x km before reaching P, and then from that point along a straight line in sea until reaching R.
Then the sea distance L is given by
L2 = (60 - x)2 + 402
L = [(60 - x)2 + 402]1/2
Total cost C = cost on land + cost in sea
C = 30000x + 50000[(60 - x)2 + 402]1/2
dC/dx = 30000 + 50000(1/2)[(60 - x)2 + 402]-1/2(2)(60 - x)(-1)
= 10000{3 - 5(60 - x)[(60 - x)2 + 402]-1/2}
For minimum, dC/dx = 0
so 3 - 5(60 - x)[(60 - x)2 + 402]-1/2 = 0
3[(60 - x)2 + 402]1/2 = 5(60-x)
9[(60 - x)2 + 402] = 25(60 - x)2
9(402) = 16(60 - x)2
900 = (60 - x)2
+/-30 = 60 - x
x = 30 or 90(rejected)
when x = 30, C = 30000(30) + 50000[900 + 1600]1/2 = 3400000
when x = 30.1 C = 30000(30.1) + 50000[894.01 + 1600]1/2 = 3400003
when x = 29.9 C = 30000(29.9) + 50000[906.01 + 1600]1/2 = 3400003
x = 30 is the minimum point
So the minimum cost is 3,400,000
