3HW7 MAXIMA-MINIMA PROBLEMS DUE: December 1/2 class period

INSTRUCTIONS. Write a complete solution to each of the following problems. Box your final answer. Use

one whole intermediate paper.

1. Find two numbers whose difference is 100 and whose product is a minimum.

2. Find two numbers whose sum is 10 and the sum of the squares is a minimum.

3. Find two positive numbers whose product is 100 and whose sum is a minimum.

4. Find the dimensions of a rectangle with area 100m2 whose perimeter is as small as possible.

5. Find two numbers whose sum is 240 and whose product is a maximum.

6. Find the dimensions of a rectangle with perimeter 240m whose area is as large as possible.

7. If one side of a rectangular field is to have a river as a natural boundary, find the dimensions of the largest rectangular field that can be enclosed by using 240m of fence for the other three sides.

8. A rectangular field is to be enclosed by a fence and then divided into two lots by another fence set at the middle. What must be the dimensions of the field with the largest area if the total length of the fencing material is 240m?

9. Suppose the cardboard is 24in by 24in, what is the maximum volume of the box constructed according to the box problem?

10. A box with a square base and open top must have a volume of 32,000 cm3. Find the dimensions of

the box that minimize the amount of material used.

11. Find the point on the line 6x + y = 9 that is closest to the point (-3, 1).

12. Show that the vertex is the point on a parabola that is closest to its focus.

13. Find the points on the ellipse 4x2 + y

2 = 4 that are farthest away from the point (1, 0).

14. What point on the hyperbola x2 – y

2 = 2 is closest to the point (0, 1)?

15. Of all the different types of isosceles triangles with fixed perimeter, which one has the greatest area?

3HW8 MAXIMA-MINIMA PROBLEMS DUE: December 5/6 class period

INSTRUCTIONS. Write a complete solution to each of the following problems. Box your final answer. Use one whole

intermediate paper.

1. Find an equation of the tangent line to the curve y = x3 – 3x2 + 5x that has the least slope.

2. A funnel of specific volume is to be in the shape of a right-circular cone. Find the ratio of the height to the base radius if the least amount of material is to be used in its manufacture.

3. Find the dimensions of the isosceles triangle of largest area that can be inscribed in a circle of radius 2 inches.

4. Find the area of the largest rectangle that can be inscribed in a right triangle with legs of lengths 3 cm and 4 cm if two sides of the rectangle lie along the legs.

5. A right circular cylinder is inscribed in a sphere of radius 1 cm. Find the largest possible surface area of such cylinder.

6. A Norman window has the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is 32 ft, find the dimensions of the window so that the window will admit the most light.

7. A paper containing 24 cm2 of printed region is to have a margin of 1.5cm at the top and bottom and 1cm at the sides. Find the dimensions of the smallest piece of paper that will fill these requirements?

8. A 5-meter wire is to be cut in two. The strength S of the wire is unit proportional to the product of the square of the one part and the cube of the other. Find the point at which this wire must be cut to maximize its strength.

9. A piece of wire 10 m long is cut into two pieces. One piece is bent into a square and the other is bent into an equilateral triangle. How should the wire be cut so that the total area enclosed is a maximum?

10. A piece of wire 10 m long is cut into two pieces. One piece is bent into a square and the other is bent into an equilateral triangle. How should the wire be cut so that the total area enclosed is a minimum?

11. A boat leaves a dock at 2:00 pm and travels due south at a speed of 20 km/h. Another boat has been heading due east at 15 km/h and reaches the same dock at 3:00 pm. At what time were the two boats closest together?

12. A telephone company has to run a line from point A on one side of a river to another point B, that is on the other side, 5 miles down the point opposite A. The river is uniformly 12 miles wide. The company can run the line along the shoreline to a point C and then run then run the line under the river to B. The cost of laying the line along the shore is P1000 per mile, and the cost of laying the line under water is twice as much. Where should point C be located to minimize the cost?

13. A direct current generator has an electromotive force of E volts and an internal resistance of r ohms, where E and r are constants. If R ohms is the external resistance, the total resistance is (r+R) ohms, and if P watts is the power, then

2

2

E RP

r R

. Show that the most power is consumed when the external resistance is equal to the internal resistance.

14. In a particular community, a certain epidemic spreads in such a way that x months after the start of the epidemic, P percent

of the population is infected, where

2

22

30xP

1 x

. In how many months will the most people be infected, and what percent

of the population is this?

15. Suppose that under a monopoly, x units are demanded daily when p pesos is the price per unit and

x =140 – p. If the total cost of producing x units is given by C(x) = x2 + 20x + 300, find the maximum daily total profit.