An o?shore oil well is located in the ocean at a point W, which is 5km from the closest shore-point A (i.e. the perpendicular distance from W to A) on a straight shoreline. The oil is to be piped to a shore-point B that is 8km from A by piping it on a straight line under water from W to some shore-point P between A and B and then on to B via a pipe along the shoreline. If the cost of laying pipe is $100; 000=km under water and $75; 000=km over land. i. Where should the point P be located, from A, to minimize the cost of laying the pipe? ii. What is the minimum cost
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A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.
XZ Plane
In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.
Euclidean Geometry
Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.
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In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.
An o?shore oil well is located in the ocean at a point W, which is 5km from the
closest shore-point A (i.e. the perpendicular distance from W to A) on a straight
shoreline. The oil is to be piped to a shore-point B that is 8km from A by piping
it on a straight line under water from W to some shore-point P between A and
B and then on to B via a pipe along the shoreline. If the cost of laying pipe is
$100; 000=km under water and $75; 000=km over land.
i. Where should the point P be located, from A, to minimize the cost of laying
the pipe?
ii. What is the minimum cost
Let the distance between A and P be x km and so distance between P and B be 8-x km
By Pythagoras theorem
Now, cost of laying the pipe is
Now , we solve the equation C'(x)=0
But x cannot be negative, so x=5.67 km
For this value of x, C'(x)>0.
Therefore, cost of laying the pipe is minimum , if the point P is 5.67 km from point A.
And minimum cost is
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