Two roads intersect at right angles. A water spring is located 65 m from one road and 55 m from the other road. A straight path is to be laid out to pass the spring from one road to the other. Find the least area that can be bounded by the roads and the path.
area in m2 =
can anyone shed some light on this questions....
Basically you want the smallest area bounded by the x- and y-axes, and any line passing through (65,55).
A line through (65,55) with slope m will have intercepts at y = 55-65m and x=65+55/m
So, the area of the enclosed triangle is
a(m) = 1/2 (55+65m)(65+55/m)
a(m) = 25/2m (11-13m)^2
da/dm = 25/2 (121/m^2 - 169)
da/dm=0 when m = ±11/13
We know we need a negative slope, so m = -11/13
The minimum area is thus 7150
oops on typos
That should be 1/2 (55-65m)(65-55/m)
To solve this problem, you can start by drawing a diagram.
Let's denote the two roads as Road A and Road B. The water spring is located 65 meters from Road A and 55 meters from Road B. To find the least area that can be bounded by the roads and the path, we need to find the dimensions of the rectangle formed by the two roads and the path.
Since the roads intersect at right angles, we can consider the distance between the roads as the length of the rectangle, and the distances from the water spring to each road as the width of the rectangle.
Let's denote the length of the rectangle as L and the width of the rectangle as W.
From the given information, we have:
Length (L) = Distance between the roads
Width (W) = Distance from the water spring to each road
To find the distance between the roads (L), we can use the Pythagorean Theorem, which states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.
Applying the Pythagorean theorem to the given information, we have:
L^2 = 65^2 + 55^2
Simplifying the equation:
L^2 = 4225 + 3025
L^2 = 7250
Taking the square root of both sides:
L = sqrt(7250)
L ≈ 85.14 meters
Therefore, the length of the rectangle is approximately 85.14 meters.
To find the width of the rectangle (W), we take the smaller of the two distances from the water spring to each road, which is 55 meters.
Now that we have the dimensions of the rectangle (L = 85.14 meters, W = 55 meters), we can find the least area bounded by the roads and the path by multiplying the length and the width:
Area = Length × Width
Area = 85.14 meters × 55 meters
Area ≈ 4683.27 square meters
Therefore, the least area that can be bounded by the roads and the path is approximately 4683.27 square meters.