A pottery factory needs to connect with the main gas pipe line. This pipeline is on the other side of a stream. A gas line must be run from the main line junction to the factory at the lowest possible cost. It costs $12.50 per foot to run pipe under water and $9.00 per foot to run it on dry land. find the exact spot across the river to pipe to and the minimum cost.

Question

The diagram shows the river to be 50 ft across forming a right triangle and the land length is 200ft

Answer 1

Another use for the Pythagorean Theorem.

a^2 + b^2 = c^2

50^2 + 200^2 = c^2
2500 + 40,000 = c^2
42,500 = c^2
206.16 = c

206.16 * $12.50 = $2,577

Or --

50(12.50) + 200(9) = 625 + 1800 = $2,425

Answer 2

It turns out that the minimum cost is when the pipe runs for 148 (approx.) ft on land, then cross the river at an angle.

The resulting cost would be about $2234.

Here's how this can be done with the help of calculus:

Assume x = the distance between the crossing point and the point directly opposite the target.

The distance on dry land is then 200-x.
Using Pythagoras Theorem, the distance under water is sqrt(x^2+50^2)

The total cost is therefore:
C(x) = 9*(200-x) + 12.5*sqrt(x^2+50^2)

Differentiate with respect to x and equate the derivative to zero for the minimum cost. Solve for x.
Confirm by the second-derivate rule or otherwise that the value of x found is a minimum (as opposed to a maximum).

Answer 3

Thank you, MathMate. My ignorance of calculus is showing. <g>

Answer 4

Not at all.

Your help in math is appreciated not only by students, but by math teachers too!

Answer 5

what a regular polygon