Find two positive numbers such that the sum of the first number and four times the second number is i00 and the product of numbers is as larger as possible. CDo not forget to use a maximum-minimum principle to justify your proper answer.)

Q: An open-top box is to be made from a 20 in by 30 in piece of cardboard by removing a square from…

A: Let each side of the square cut-off from each corner be x in. then the length of the base of the box…

Q: Find a pair of non-negative numbers that have a product of 162 and minimize the sum of two times the…

A: Our Aim is to find a pair of non-negative numbers that have a productof 162 and minimizes the sum of…

Q: 2. What is the minimum product of two numbers that differ by 15? What are the two numbers?

A: Let one number be x and another number be x+15 So their product=x(x+15)=x2 + 15x =f(x) (say)

Q: 3. A piece of wire 36 in. is cut into two parts, one of which is bent into the shape of an…

A:

Q: 3.3.61 Among all pairs of numbers whose difference is 20, find a pair whose product is as small as…

A: let two number is x and y Their difference is 20

Q: 12. Of all numbers whose sum is 70, find the two that have the maximum product.

A: let x and y be two numbers whose sum is 70   we have to maximize  M=xy where x+y=70

Q: A wire of length 10 is to be cut into two pieces, one being bent to form a square and the other to…

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Q: A piece of 1m wire is cut into two parts, one bent into the shape of a circle and the other into the…

A:

Q: Show that the interval [0,1) does not have a maximum element; that is, there is no largest number…

A:

Q: 3. A farmer has 6000 m of fencing and wishes to create a rectangular field subdivided into four…

A:

Q: 4) A farmer has 2000 feet of fence and wants to build a rectangular enclosure along a straight wall.…

A: Given:-rectangular fence along a straight wallthere is no fence along the walltotal length of…

Q: State Pontyagin's maximum principle

A: Pontryagin's maximum principle is used in optimal control theory to find the best possible control…

Q: Two nonnegative numbers whose product is 36a? (where a > 0) and whose sum is a minimum, are

A: Now we will solve the problem.

Q: What is the minimum value of x^2+100/x on a horizontal span of 1 to 5?

A: The given expression is x2+100x;1,2,3,4,5. Let us use the graph to analyze the above function for…

Q: 11. Of all numbers whose sum is 50, find the two that have the maximum product.

A: Let x and y be the two numbers.then,x+y = 50 y = 50-x            ..........(1)Let P be their product…

Q: A rectangular garden is to be constructed using a rock wall as one side of the g and wire fencing…

A: We are asked to find dimensions of the rectangle.

Q: A candy box is made from a piece of cardboard that measures 45 by 24 inches. Squa

A: Given: A candy box is made from a piece of cardboard that measures  45 by 24 inches. Squares of…

Q: Find the dimensions of a rectangular box of maximum volume such that the sum of the lengths of its…

A: According to the question, we have a rectangular box and the sum of the lengths of its 12 edges is…

Q: 1. A rectangular lot is to be fenced on the sides and front, and is to be divided into two lots with…

A:

Q: An open box is to be made from a square piece of material, 12 inches, by cutting equal squares from…

A: Here we use the concept of APPLICATIONS OF DERIVATIVES

Q: 13. Of all numbers whose difference is 6, find the two that have the minimum product.

A: Let x and y be such numbers. x-y=6 P=xy. Substitute x=6+y in P=xy and obtain that P=y6+y=6y+y2.

Q: We want to construct a box with a square base and we only have 10 m² of material to use in…

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Q: What is the absolute minimum value of p(x)=2x2-x-2 over [−2,6]?

A:

Q: An open box will be made by cutting a square from each corner of a 10-cm by 16-cm piece of cardboard…

A: To find: The size of square to be cut from the corner of cardboard to maximize volume and also find…

Q: (2) (a) For which PDE that we've studied is the Maximum Principle true? Describe the Maximum…

A:

Q: Among all pairs of numbers whose difference is 10, find a pair whose product is as small as…

A:

Q: A package in the shape of a rectangular box can be sent by post as a postal order if the sum of its…

A: Solution by using Lagrange multipliers as follows :

Q: A rectangular field is to be covered by 800-meter of hog wire in such a way that one side has a…

A: The Perimeter of rectangular field = 800 m Let length of field = x m Width of field = y m

Q: Find two positive numbers whose product is 50 such that the sum of the first plus 2 times the second…

A: Given, Product of 2 positive number is 50  First number+2(second number)=S, is a minimum.

Q: 6. A square sheet of cardboard of side 12 in. square is used to make an open box by cutting out…

A:

Q: A container with an open top to have 10m^3 capacity and be made of thin sheet metal. Calculate the…

A: Let A be the total area of metal used to make the box, and let x and y be the length and width and z…

Q: A metal wire of length 80 cm is cut into two parts such that one part is 4k cm long. Each part of…

A:

Q: A farmer wants to enclose a rectangular field with area 1,000 square meters by fencing it into four…

A:

Q: A piece of wire of length 1 meter is cut into two parts, one of which is bent into the shape of a…

A: A piece of wire of length 1 meter is cut into two parts , one of which is bent the shape of a circle…

Q: A string of length L is cut into arbitrarily many pieces. Some or all pieces are made to circles and…

A: Let the string is cut into n equal pieces. Then the length of each piece will be Ln. Now, let x…

Q: A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal…

A: This question is based on application of derivative.

Q: 8. What is the maximum/minimum value? *

A: 8. To find the maximum/minimum value9.To find  the x-interecepts of fx10. Does fx opens upwards or…

Q: A piece of wire 10 m long is cut into two pieces. One pieceis bent into a square and the other is…

A: To find: The way to use the wire such that the area enclosed by the triangle and square would be…

Q: A courier service will deliver a package only if the length plus girth does not exceed 69 inches.…

A: Given that, A courier service will deliver a package only if the length plus girth does not exceed…

Q: A rectangular box without a lid is to be made from 12 m2 of cardboard. Find the maximum volume of…

A: Given that a rectangular box without a lid is to be made from 12 m² of cardboard.

Q: Find the rectangle of the maximum area that can be inscribed in a right triangle with legs of length…

A:

Q: x and y are both positive numbers such that P=xy3. If the sum of both numbers is 24, determine x and…

A: given data p = xy3x+y =24

Q: A open-top box is to be made from a 24 in by 36 in piece of cardboard by removing a square from each…

A:

Q: Find two positive numbers such that their product is a maximum subject to the condition that the sum…

A:

Q: On the domain (0, o), find the minimum value of f(x) = D %3D

A: Given function

Q: is the total cost of production for a 7-hour day, where x machines are used to produce jackets and y…

A: The Extreme value theorem states that if a function is continuous on a closed interval a, b, then…

Q: A printed page is to contain 60 cm2 of printed material (not including the margins) with margins (on…

A:

Q: 11. Determine three positive numbers whose sum is 21, so that your product is the maximum possible…

A:

Q: Find the dimensions of a rectangular box of maximum volume such that the sum of the lengths of its…

A: To find: Dimension of a rectangular box. Given: Volume of the box is maximum and sum of edges of box…