Example of optimization , Mathematics

Assignment Help:

A piece of pipe is carried down a hallway i.e 10 feet wide.  At the ending of the hallway the there is a right-angled turn & the hallway narrows down to 8 feet wide. What is the longest pipe which can be carried (always keeping it horizontal) around the turn in the hallway?

Solution

Let's begin with a sketch of the situation therefore we can obtain a grip on what's going on and how we will solve this.

345_tanglent1.png

The largest pipe which can go around the turn will do therefore in the position illustrates above.  One end will be touching the outer wall of the hall way at A & C and the pipe will contact the inner corner at B. Let's suppose that the length of the pipe in the little hallway is Lwhile L2  is the length of the pipe into the large hallway. Then the pipe has a length of L = L1 + L2 .

Now, if θ = 0 then the pipe is totally in the wider hallway and we can illustrates that as θ → 0

54_tanglent.png

then L → ∞ .  Similarly, if θ = ∏/2 the pipe is totally in the narrow hallway and as θ → ∏/2   we also have L → ∞ .  Therefore, somewhere in the interval 0 < θ < ∏/2    is an angle that will minimize L and oddly sufficient i.e. the length that we're after. The largest pipe which will fit around the turn will actually be the minimum value of L.

The constraint for this problem is not so obvious and there are in fact two of them.  The constraints for this difficulty are the widths of the hallways.  We'll utilize these to obtain an equation for L in terms of θ & then we'll minimize this new equation.

Therefore, by using basic right triangle trig we can illustrates that,

L1 = 8 sec θ           L2  = 10 csc θ        ⇒       L = 8 sec θ + 10 csc θ

Therefore, differentiating L gives,

                           L′ = 8 sec θ tan θ -10 csc θ cot θ

Setting this equivalent to zero and solving out specified,

                    8 sec θ tan θ = 10 csc θ cot θ

sec θ tan θ /csc θ cot θ = 10/8

sin θ tan2 θ /cos θ =5/4           ⇒         tan3 θ = 1.25

Solving for θ gives,

Therefore, if θ = 0.8226 radians then the pipe will contain a minimum length and will just fit around the turn. Anything larger will not fit about the turn that's why the largest pipe that can be carried around the turn is,

                              L = 8 sec (0.8226 ) + 10 csc (0.8226) = 25.4033 feet


Related Discussions:- Example of optimization

Trigonometry, what are reason inside a circle?

what are reason inside a circle?

Find the number of students side of the square, A teacher on attempting to ...

A teacher on attempting to arrange the students for mass drill in the form of a solid square found that 24 students were left over. When he increased the size of the square by one

Geometyr, Lines EF and GH are graphed on this coordinate plane. Which point...

Lines EF and GH are graphed on this coordinate plane. Which point is the intersection of lines EF and GH?

TRIANGLES, ABCD is a trapezium AB parallel to DC prove square of AC - squar...

ABCD is a trapezium AB parallel to DC prove square of AC - square of BCC= AB*

Basic operations for complex numbers, Now we have to discuss the basic oper...

Now we have to discuss the basic operations for complex numbers. We'll begin with addition & subtraction. The simplest way to think of adding and/or subtracting complex numbers is

Changing the base of the index, Changing The Base Of The Index For com...

Changing The Base Of The Index For comparison reasons if two series have different base years, this is difficult to compare them directly. In such cases, it is essential to ch

SURFACE AREA AND VOLUMES, Metallic spheres of radii 6 centimetre, 8 centime...

Metallic spheres of radii 6 centimetre, 8 centimetre and 10 centimetres respectively are melted to form a single solid sphere. Find the radius of the resulting sphere.

Marginal probability, Marginal Probability Probability of event A happe...

Marginal Probability Probability of event A happening, denoted by P(A), is called single probability, marginal or unconditional probability. Marginal or Uncondi

Applying quadratics math question, A boat tour company charges $11 for a ha...

A boat tour company charges $11 for a harbour tour and averages 450 passengers on Saturdays. Over the past few months, the company has been experimenting with the price of a tour a

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd