Problems

1             Split 20 into the sum of two numbers x ≥ 0 and y ≥ 0 such that the product of x and y ² is a maximum.

2            Find two numbers x ≥ 0 and y ≥ 0 such that x + y = 8 and x² + y² is a minimum.

3            Find two numbers x ≥ 1 and y ≥ 1 such that xy = 50 and 2x + y is a maximum.

4            Find the rectangle with perimeter 8 which has maximum area.

5            Find the maximum value of x³y if x and y belong to [0,1] and x + y = 1.

6            A rectangular box which is open at the top can be made from a 10 by 12 inch piece of metal by cutting a square from each corner and bending up the sides. Find the dimensions of the box with greatest volume.

7            A poster of total area 400 sq in. is to have a margin of 4 in. at the top and bottom and 3 in. at each side. Find the dimensions which give the largest printed area.

8            A man can travel 5 mph along the path AB and 3 mph off the path as shown in the figure. Find the quickest route APC from the point A to the point C.

9            Find the dimensions of the right triangle of maximum area whose hypotenuse has length one.

10            Find the dimensions of the isosceles triangle of maximum area which has perimeter 3.

11             Find the five-sided figure of maximum area which has the shape of a square topped by an isosceles triangle, and such that the sum of the height of the figure and the perimeter of the square is 20 ft.

12            A wire of length L is to be divided into two parts; one part will be bent into a square and the other into a circle. How should the wire be divided to make the sum of the areas of the square and circle as large as possible? As small as possible?

13            Find the area of the largest rectangle which can be inscribed in a semicircle of radius r.

14            Find the dimensions of the rectangle of maximum area which can be inscribed in an equilateral triangle as shown in the figure.

15            Find the shape of the right circular cylinder of maximum volume which can be inscribed in a right circular cone of height 3 and base of radius 1.

16            Find the shape of the right circular cone of maximum volume which can be inscribed in a given sphere.

17            Find the shape of the cylinder of maximum volume such that the sum of the height and the circumference of the base is equal to 4.

18            Find the shape of the largest trapezoid which can be inscribed in a semicircle as shown in the figure.

19            If a farmer plants x units of wheat in his field, 0 ≤ x ≤ 100, the yield will be 10x - x²/10 units. How much wheat should he plant for the maximum yield?

20            In Problem 19 above, it costs the farmer $100 for each unit of wheat he plants, and he is able to sell each unit he harvests for $50. How much should he plant to maximize his profit?

21            A professional football team has a stadium which seats 60,000. It is found that x tickets can be sold at a price of p = 10 - x/10,000 dollars per ticket. Find the values of x and p at which the total money received will be a maximum.

22            In Problem 21 a tax of $1 per ticket is added onto the price. Find x and p so that the total revenue after taxes is a maximum.

23            A store can buy up to 300 seconds of advertising time daily on the radio at the rate of $2/sec for the first 100 sec, and $1/sec thereafter, x seconds on the radio increases daily sales by 32√x dollars. How many seconds on the radio will yield the maximum profit?

24            Work Problem 23 if the cost of advertising time is $1/sec for the first 100 sec and $2/sec thereafter.

25            Find the real number which most exceeds its square.

26            Find the rectangle of area 9 which has the smallest perimeter.

27            Find the right triangle of smallest area in which a 1 by 2 rectangle can be inscribed as shown in the figure.

28            A farmer wishes to enclose 10,000 sq ft of land along a river by three sides of fence as shown in the figure. Find the dimensions which require the minimum length of fence.

29            Find the shortest distance between the line y = 1 - 4x and the origin.

30            Find the shortest distance between the curve y = 2/x and the origin.

31            A warehouse is to be built in the shape of a rectangular solid with a square base. The cost of the roof per unit area is three times the cost of the walls. Find the shape which will enclose the maximum volume for a given cost.

32            A rectangular box with volume 1 cu ft is to be made with a square base and no top. Find the dimensions which require the smallest amount of material.

33            Find the dimensions of the right circular cylinder of volume 1 cu ft which has the smallest surface area (top plus bottom plus sides).

34            Find the dimensions of the right circular cone of smallest volume which can be circumscribed about a sphere of radius r.

35            Given two real numbers a and b, find x such that (x - a)² + (x - b)² is a minimum.

36            The area of a sector of a circle with radius r and central angle θ is A = ½r²θ, and its arc has length s = rθ. Find r and θ so that 0 < θ < 2π, the sector has area 1, and the perimeter is a minimum.

37            Show that among all right circular cylinders of volume 1 cu ft which are open at both ends, there is no maximum or minimum surface area.

38            The population of a country at time t = 0 is 50 million and is increasing at the rate of one million people per year. The national income at time t is (20,000 + t²) million dollars per year. At what time t ≥ 0 is the per capita income (= national income ÷ population) a minimum?

39            A man estimates that he can paint his house in x hours of his spare time if he buys equipment costing 200 + 2000/x² dollars, and that his spare time is worth $2/hr. How many hours should he take?

40            An artisan can produce x items at a total cost of 100 + 5x dollars and sell x items at a price of 10 - x/100 dollars per item. Find the value of x which gives the maximum profit.

41            A manufacturer can produce any number of buttons at a cost of two cents per button and can sell x buttons at a price of 1000/√x cents per button. How many buttons should be produced for maximum profit ?