In order to deliver products to market, a factory which
is 20 km away from the railway needs to ship its merchandise to a new
railway link D from the highway and then to B railroad. It is known that
the total length of the railroad is 100 km and the rate of the unit price
to transport through railroad to freeway is 3:5. Where should D be located
so that the manufacturer pays the lowest cost for shipping?
Let the distance between A and D be x km, then the length of DB equals
100 - x km and CD equals (x2 + 202)0.5 km.
It is given that the rate of the unit price to ship through railway
to road is 3:5, thus the shipping cost for the railroad and highway per
kilometer can be expressed as 3k and 5k respectively.
Let the total payment for the transportation be y.
The total transportation fee is the sum of the railway and freeway payment.
y = CD(5k) + DB(3k)
Substituting DB = 100 - x and CD = (x2 + 202)0.5 into
the total payment equation gives
y = 5k(x2 + 202)0.5 +
3k(100 - x)
Recall that the minimum payment may occur at the point where the derivative
of the function is 0 or does not exist (critical
point). In order to find these points, the derivative of the function
needs to be calculated.
As mentioned previously dy/dx needs to be 0 or does not exist. The case, dy/dx
does not exist, cannot happen because no matter what value x is,
the term (x2 + 202)0.5 cannot be 0.
The case, dy/dx equals 0, makes to
be 0. Since k is a nonzero positive coefficient, x can be found by
solving the equation .
Squaring both sides of the equation gives
Simplifying this equation gives
16x2 = 9(400)
x = 15 or x = -15