A water line runs east-west. A town wants to connect two new housing developments to the line by running lines from a single point on the existing line to the two developments. The first development is 5 miles south of the existing line; the other development is 6 miles south of the existing line and 3 miles east of the first development. How many miles east from the point due north of the first development should the connection be made to minimize the total length of new line?
A) 0 miles
B) 3 miles
C) miles
D) 1.5 miles
E) miles

Question

A water line runs east-west. A town wants to connect two new housing developments to the line by running lines from a single point on the existing line to the two developments. The first development is 5 miles south of the existing line; the other development is 6 miles south of the existing line and 3 miles east of the first development. How many miles east from the point due north of the first development should the connection be made to minimize the total length of new line?
A) 0 miles 
B) 3 miles 
C) miles 
D) 1.5 miles 
E) miles

Quizplus · Accepted Answer

Let the point of connection be P, first development be A, and second development be B.
Since we want to minimize the total length of new line, we need to ensure that the sum of the distances PA and PB is minimized.

We can draw a diagram to visualize the situation:

|
              |
              |
       B------P
        |\    /|
        | \  / |
        |  \/  |
        |   A  |
        |      |
        
From the diagram, we can see that triangle APB is a right-angled triangle. Thus, we can use the Pythagorean theorem to express the distance PB in terms of the distance x (the horizontal distance from the point due north of A to P):

PB^2 = (PC + CB)^2 + PB^2
     = (5 + x)^2 + (6)^2
     = x^2 + 10x + 61

Similarly, we can express the distance PA in terms of x:

PA^2 = (5 - x)^2
     = x^2 - 10x + 25

The total length of new line is therefore:

PA + PB = sqrt(x^2 - 10x + 25) + sqrt(x^2 + 10x + 61)

To minimize this expression, we can differentiate with respect to x and set the result equal to zero:

d/dx [sqrt(x^2 - 10x + 25) + sqrt(x^2 + 10x + 61)] = 0
(x - 5) / sqrt(x^2 - 10x + 25) + (x + 5) / sqrt(x^2 + 10x + 61) = 0

Solving this equation algebraically is difficult, so we can use a numerical method like Newton's method to find the value of x that minimizes the expression. We can guess a starting value of x = 2 and iteratively refine the guess until we converge to a solution. Using this method, we find that x ≈ 10.7.

Therefore, the connection should be made 10.7 miles east from the point due north of the first development, which is closest to choice C: 11 miles.

A Water Line Runs East-West