maximum And Minimum Problems

[سامح الدهشان]

السلام عليكم ورحمة الله وبركاته

الأخوة الافاضل :

تمارين علي القيم القصوي المطلقة

حتي يستفيد الجميع

[سامح الدهشان]

example

Find two nonnegative numbers whose sum is 9 and so that the product of one number and the square of the other number is a maximum.

أوجد العددين ( الغير سالبين ) والذي مجموعهما = 9

وكذلك حاصل ضرب أحدهما في مربع الأخر أكبر مايمكن ؟

[سامح الدهشان]

An open rectangular box with square base is to be made from 48 ft.2 of material. What dimensions will result in a box with the largest possible volume ?

صندوق علي شكل متوازي مستطيلات مفتوح من أعلي

قاعدته مربعة الشكل كمية المادة المعدنية ( الصفائح المعدنية ) المستخدمه في صنعة

= 48 قدم مربع أوجد أبعاد الصندوق التي تؤدي الي أكبر حجم ممكن

( أو التي تجعل الحجم أكبر مايمكن ) مع أيجاد قيمة الحجم عندئذا ً

[سامح الدهشان]

أيضا الملاحظات المتبعة عند حل تمارين علي القيم القصوي


GUIDELINES FOR SOLVING MAX./MIN. PROBLEMS

1. Read each problem slowly and carefully. Read the problem at least three times before trying to solve it. Sometimes words can be ambiguous. It is imperative to know exactly what the problem is asking. If you misread the problem or hurry through it, you have NO chance of solving it correctly.
2. If appropriate, draw a sketch or diagram of the problem to be solved. Pictures are a great help in organizing and sorting out your thoughts.

3. Define variables to be used and carefully label your picture or diagram with these variables. This step is very important because it leads directly or indirectly to the creation of mathematical equations.

4. Write down all equations which are related to your problem or diagram. Clearly denote that equation which you are asked to maximize or minimize. Experience will show you that MOST optimization problems will begin with two equations. One equation is a "constraint" equation and the other is the "optimization" equation. The "constraint" equation is used to solve for one of the variables. This is then substituted into the "optimization" equation before differentiation occurs. Some problems may have NO constraint equation. Some problems may have two or more constraint equations.

5. Before differentiating, make sure that the optimization equation is a function of only one variable. Then differentiate using the well-known rules of differentiation.

6. Verify that your result is a maximum or minimum value using the first or second derivative test for extrema.

[سامح الدهشان]

. Read each problem slowly and carefully. Read the problem at least three times before trying to solve it

أقرأ التمرين ببطء وعناية ( أهتمام ) ==> بتأني

أقرأ التمرين علي الأقل ثلاث مرات قبل محاولة الحـــــــــل


. Sometimes words can be ambiguous. It is imperative to know exactly what the problem is asking

أحيانا ً بعض الكلمات تكون مبهمة ( غير وأضحة) ولاتعرف من خلالها ما هو المطلوب بالضبط


If you misread the problem or hurry through it, you have NO chance of solving it correctly.
2. If appropriate, draw a sketch or diagram of the problem to be solved. Pictures are a great help in organizing and sorting out your thoughts إن قرأت التمرين بسرعة دون دقة قد تفقد فرصة الحل بالشكل السليم

ولذلك ينبغي عليك تحليل ورسم التمرين عند حله لان الصور تساعد بشكل كبير

في فهمك للمعطيات وتحليلك للمطلوب


Define variables to be used and carefully label your picture or diagram with these variables. This step is very important because it leads directly or indirectly to the creation of mathematical equations.

عرف المتغيرات التي سوف تستخدم بعناية علي الصورة المرسومة

هذة الخطوة مهمة جدا ً لان ذلك سيؤدي مباشرة وغير مباشر لتكوين المعادلات الرياضية

[سامح الدهشان]

. Write down all equations which are related to your problem or diagram. Clearly denote that equation which you are asked to maximize or minimize. Experience will show you that MOST optimization problems will begin with two equations. One equation is a "constraint" equation and the other is the "optimization" equation. The "constraint" equation is used to solve for one of the variables. This is then substituted into the "optimization" equation before differentiation occurs. Some problems may have NO constraint equation. Some problems may have two or more constraint equations.

أكتب أسفل الصورة ( الرسمة ) جميع المعادلات الخاصة بالشكل

من حيث محيط القاعدة أو مساحة القاعدة أو الأرتفاع

أو المساحة الجانبية أو المساحة الكلية أو الأرتفاع

من المعطيات سنحصل علي معادلة بين متغيرين نوجد أحدهما بدلالة الأخر

من المطلوب : سنكون الدالة المراد التعامل معها ونحدد المطلوب

أنتبه يتم التعامل مع الدالة بعد وضعها في أبسط صورة ممكنة لاتبدأ بالإشتقاق

الإ أذا كانت الدالة في أبسط صورة وفي متغير وأحد

[سامح الدهشان]

. Before differentiating, make sure that the optimization equation is a function of only one variable. Then differentiate using the well-known rules of differentiation

قبل الإشتقاق : تأكد من الدالة المطلوبة في متغير وأحد

ثم أستخدم قواعد الإشتقاق علي حسب شكل الدالة


. Verify that your result is a maximum or minimum value using the first or second derivative test for extrema

للتحقق من أن القيمة الناتجة تجعل المطلوب اكبر مايمكن أو أصغر مايمكن

أستخدم أختيار المشتقة الاولي أو أختبار المشتقة الثانية

[سامح الدهشان]

والأن مع مزيد من التمارين :

Build a rectangular pen with three parallel partitions using 500 feet of fencing. What dimensions will maximize the total area of the pen ?

0000000000000000000000000000000000000000000000000000

A container in the shape of a right circular cylinder with no top has surface area 3 ft.2 What height h and base radius r will maximize the volume of the
cylinder ?

صفيحه معدنية علي شكل أسطوانة دائرية قائمة مفتوحه من أعلي مساحتها السطحية = 3 قدم مربع

وأرتفاعها ( h ) وطول نصف قطر قاعدتها ( r ) ماهي الابعاد التي تجعل حجم الأسطوانة

أكبر مايمكن
000000000000000000000000000000000000000000000000000

A sheet of cardboard 3 ft. by 4 ft. will be made into a box by cutting equal-sized squares from each corner and folding up the four edges. What will be the dimensions of the box with largest volume ?

قطعة من الورق المقوي طولها 4 قدم وعرضها 3 قدم صنعت منها صندوق

وذلك بقطع مربعات متساوية من أركانها الأربعة وثنيت الأجزاء البارزة منها

ماهي أبعاد الصندوق التي تجعل الحجم أكبر مايمكن

[سامح الدهشان]

Consider all triangles formed by lines passing through the point (8/9, 3) and both the x- and y-axes. Find the dimensions of the triangle with the shortest hypotenuse.

00000000000000000000000000000000000000000000000

Find the point (x, y) on the graph y = sqr(x) of nearest the point (4, 0).



0000000000000000000000000000000000000000000000000
A cylindrical can is to hold 20 m.3 The material for the top and bottom costs $10/m.2 and material for the side costs $8/m.2 Find the radius r and height h of the most economical can.

ملحوظة ( 20 باي ) أو ( 0 2 ط ) مترمكعب

[سامح الدهشان]

You are standing at the edge of a slow-moving river which is one mile wide and wish to return to your campground on the opposite side of the river. You can swim at 2 mph and walk at 3 mph. You must first swim across the river to any point on the opposite bank. From there walk to the campground, which is one mile from the point directly across the river from where you start your swim. What route will take the least amount of time ?

000000000000000000000000000000000000000000000000

Construct a window in the shape of a semi-circle over a rectangle. If the distance around the outside of the window is 12 feet, what dimensions will result in the rectangle having largest possible area ?

00000000000000000000000000000000000000000000000000

There are 50 apple trees in an orchard. Each tree produces 800 apples. For each additional tree planted in the orchard, the output per tree drops by 10 apples. How many trees should be added to the existing orchard in order to maximize the total output of trees ?

[سامح الدهشان]

Find the dimensions of the rectangle of largest area which can be inscribed in the closed region bounded by the x-axis, y-axis, and graph of y=8-x3 . (See diagram.)

0000000000000000000000000000000000000000000000000000

Consider a rectangle of perimeter 12 inches. Form a cylinder by revolving this rectangle about one of its edges. What dimensions of the rectangle will result in a cylinder of maximum volume ?

000000000000000000000000000000000000000000000000

A movie screen on a wall is 20 feet high and 10 feet above the floor. At what distance x from the front of the room should you position yourself so that the viewing angle of the movie screen is as large as possible ? (See diagram.)

http://www.al3ez.net/upload/c/sameh_MaxMin14A.gif

[سامح الدهشان]

Find the dimensions (radius r and height h) of the cone of maximum volume which can be inscribed in a sphere of radius 2.

00000000000000000000000000000000000000000000000

What angle between two edges of length 3 will result in an isosceles triangle with the largest area ? (See diagram.)

http://www.al3ez.net/upload/c/sameh_MaxMin16A.gif

[سامح الدهشان]

Of all lines tangent to the graph of y = 6 /(x^2+3), find the tangent lines of mimimum slope and maximum slope.

00000000000000000000000000000000000000000000000

Find the length of the shortest ladder that will reach over an 8-ft. high fence to a large wall which is 3 ft. behind the fence. (See diagram.)

http://www.al3ez.net/upload/c/sameh_MaxMin18A.gif

[سامح الدهشان]

Find the point P = (x, 0) on the x-axis which minimizes the sum of the squares of the distances from P to (0, 0) and from P to (3, 2).


0000000000000000000000000000000000000000000000000

Car B is 30 miles directly east of Car A and begins moving west at 90 mph. At the same moment car A begins moving north at 60 mph. What will be the minimum distance between the cars and at what time t does the minimum distance occur ?

0000000000000000000000000000000000000000000000000

A rectangular piece of paper is 12 inches high and six inches wide. The lower right-hand corner is folded over so as to reach the leftmost edge of the paper (See diagram.).

http://www.al3ez.net/upload/c/sameh_MaxMin21A.gif

[أحمد سعد الدين]

السلام عليكم ورحمة الله وبركاته

جزاك الله خيرا الأستاذ سامح

تمارين جميلة ومتنوعة

وسأحاول حل بعض التمارين - ولكن ليس بالترتيب - مع ذكر نص التمرين



What angle between two edges of length 3 will result in an isosceles triangle with the largest area ? (See diagram.)

http://up.arabsgate.com/u/1524/3443/43742.jpg

[أحمد سعد الدين]

Find the length of the shortest ladder that will reach over an 8-ft. high fence to a large wall which is 3 ft. behind the fence. (See diagram.)


http://up.arabsgate.com/u/1524/3443/43754.gif

[أحمد سعد الدين]

A rectangular piece of paper is 12 inches high and six inches wide. The lower right-hand corner is folded over so as to reach the leftmost edge of the paper (See diagram.).


http://up.arabsgate.com/u/1524/3443/43755.jpg

[سامح الدهشان]

جزاك الله خيرا ً والدنا الفاضل

الأستاذ : أحمد سعد الدين علي هذا المجهود

الأكثر من رائع والحلول الرائعة وأسمح لي بتغيير العنوان

ليتفق مع المضمون

[أحمد سعد الدين]

Car B is 30 miles directly east of Car A and begins moving west at 90 mph. At the same moment car A begins moving north at 60 mph. What will be the minimum distance between the cars and at what time t does the minimum distance occur ?



http://up.arabsgate.com/u/1524/3443/43765.jpg

[أحمد سعد الدين]

Find the point P = (x, 0) on the x-axis which minimizes the sum of the squares of the distances from P to (0, 0) and from P to (3, 2).


http://up.arabsgate.com/u/1524/3443/43771.jpg