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.
The diagram shows the river to be 50 ft across forming a right triangle and the land length is 200ft
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
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).
Thank you, MathMate. My ignorance of calculus is showing. <g>
Not at all.
Your help in math is appreciated not only by students, but by math teachers too!
what a regular polygon
To find the exact spot across the river to pipe to, and the minimum cost, we need to consider the cost of running the pipe under the water and on dry land.
Let's analyze the situation:
1. We have a right triangle, where the river is the hypotenuse, and the land length is one of the legs. The other leg represents the length of the pipe needed to be run under the water.
2. The cost of running the pipe under water is $12.50 per foot, while the cost of running it on dry land is $9.00 per foot.
Now, let's calculate the length of the pipe needed to run under the water:
Using the Pythagorean theorem, we know that the square of the length of the hypotenuse (river) is equal to the sum of the squares of the other two sides (legs). So, in this case, we have:
river^2 = land^2 + underwater^2
Substituting the given values:
50^2 = 200^2 + underwater^2
2500 = 40000 + underwater^2
Underwater^2 = 2500 - 40000
Underwater^2 = -37500
Since the result is negative, this means there is no real solution for the length of the pipe needed to run under the water. Therefore, we can conclude that the pipeline should be run entirely on dry land, from the main line junction to the factory.
Now, let's calculate the length of the pipeline:
The length of the pipeline is equal to the length of the land. Given that the land length is 200 ft, the pipeline should be 200 ft long.
Finally, let's calculate the minimum cost:
Since the pipeline runs entirely on dry land, the cost for the entire length is $9.00 per foot. Therefore, the minimum cost is:
Minimum cost = Length of the pipeline * Cost per foot
Minimum cost = 200 * $9.00
Minimum cost = $1800
Therefore, the exact spot across the river to pipe to is not required, as the pipeline should be run entirely on dry land. The minimum cost for the construction of the pipeline is $1800.