Solve the given problems. A water pumping station is to be...

Solve the given problems. A water pumping station is to be built on a river at point $P$ in order to deliver water to points $A$ and $B$. See Fig. $2.49 .$ The design requires that $\angle A P D=\angle B P C$ so that the total length of piping that will be needed is a minimum. Find this minimum length of pipe. (FIGURE CAN'T COPY)

Solve the given problems.
A water pumping station is to be built on a river at point $P$ in order to deliver water to points $A$ and $B$. See Fig. $2.49 .$ The design requires that $\angle A P D=\angle B P C$ so that the total length of piping that will be needed is a minimum. Find this minimum length of pipe.
(FIGURE CAN'T COPY)

Key Concepts

Reflection Principle

This concept involves reflecting a point or a path across a barrier (or line) to transform a problem with a change in direction into one with a straight-line path. In many minimum distance problems, applying the reflection method simplifies the process by allowing one to use the property that the angle of incidence equals the angle of reflection, thereby converting a segmented path into a single straight line that minimizes the distance.

Geometric Optimization

Geometric optimization focuses on finding the best solution (such as the shortest or most efficient path) within a geometric framework. This involves applying principles and transformations from geometry, such as reflecting points or leveraging symmetry, to satisfy constraints (like equal angles) and minimize a total distance or cost.

Fermat’s Principle

Originally a physical principle used to determine the path of light, Fermat’s principle states that the path taken between two points is the one that minimizes travel time. This principle is analogous to many geometric minimization problems where the optimal path also minimizes the distance, often leading to the condition of equal angles at points of change in direction.

Transcript

0:00 Welcome to numerate.

00:01 In the current problem we are given that a water pumping station is to be built on a river at point p.

00:08 So this is where a station will be built.

00:13 And from p we have to leave water to two locations a and b.

00:20 And this is also such that this angle and this angle is same.

00:25 Okay, angle b, p, c and a p d should be the same.

00:30 So now it is also given the length, the distance between them is 12.

00:36 Then what will be the length of the types? that is b, p and a.

00:41 So we want to find how much is b -p and how much is p.

00:48 Now consider triangles b -c -p and a -d -p.

00:53 Triangle p c p and triangle a d p okay angle b c p is equal to angle a d p is equal to angle a d p which is is equal to 90 degree the right angles angle b p c is equal to angle a p d correct why? because it is given.

01:33 This is one of the given condition.

01:38 So, of course, the remaining angles will be the same, which implies triangle b, c, p, similar to triangle, a, d.

01:53 Okay.

01:55 Now, it is also given that this length is 6, this length is 10, this total length is 12, and we got to find how much is this okay how much will be the hypoteness of both the pipes now note that whatever is the ring length okay they should be in equal ratio if this is 6 is to 10 then even for cp is to p d will be the same so this implies bc by cp is is equals to ad by cd, sorry, pd.

02:43 Okay, now what is cp? cp is nothing but cd minus pd, correct? from cd, if i subtract pd that will be cp.

02:55 So i can write bc divided by 12 minus pt equals to a d divided by pt correct now let us substitute p c is six divided by 12 minus pd is equals to 10 divided by pd so if i substitute or divide by two this will become three this will become five so therefore what we can write we can write three times pd is equals to 5 times 12 minus pd all right so we have three times pd is equals to 60 minus 5 pd which is is equal to 8 into pd is equals to 60 so pd is equals to 60 so pd is equals to 60 by 8 if i simplify it by dividing with 2, we get 30 and then we get 4.

04:14 Again if i divide it gets 2 and it becomes 15.

04:17 So final answer will be 7 .5.

04:21 Okay, that means what? p .d.

04:24 This length is 7 .5.

04:27 So what will be length of ap? ap square will be using pythalysis theorem will be...

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