Asked by Ashley
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....
area in m2 =
can anyone shed some light on this questions....
Answers
Answered by
Steve
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
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
Answered by
Steve
oops on typos
That should be 1/2 (55-65m)(65-55/m)
That should be 1/2 (55-65m)(65-55/m)
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