1. ## Calculus.....

A cable is to be run from a power plant on one side of a
river to a factory on the other side. It costs $4 per meter to run the cable over land, while it costs$5 per
meter to run the cable under water. Suppose the river
is 300 meters wide and the factory is 1000 meters
downstream from the power plant.
What is the most economical route to lay the cable?
How much will it cost?

2. Originally Posted by spencersin
A cable is to be run from a power plant on one side of a
river to a factory on the other side. It costs $4 per meter to run the cable over land, while it costs$5 per
meter to run the cable under water. Suppose the river
is 300 meters wide and the factory is 1000 meters
downstream from the power plant.
What is the most economical route to lay the cable?
How much will it cost?
Always draw a diagram (sorry I used ft instead of meters )

From the diagram we can find the cost function by multiplying the distance by cost per meter

$\displaystyle c(x)=4(1000-x)+5\sqrt{(300)^2+x^2}$

Now we take the derivative to get

$\displaystyle \frac{dc}{dx}=-4+\frac{5x}{\sqrt{(300)^2+x^2}}$

Now we set this equal to zero and solve for x.