4.4 Optimization Problems

1. Determine where the point between and should be located to maximize the angle .

Let , , .

Then

. Solving , we get

. Since we have

.

2. A shop sells 200 MP3 players per week while each costs . According to the market research, sales will increase by 20 MP3 players per week for discount. How much should they discount to maximize profits?

If discount,

profits is maximum when .

They should discount $120 or $130.

3. The height of a safe is meters and its bottom is in the shape of a square whose side is meters. It costs Korean won per to make the top and the bottom, and Korean won per to make the side. Find the maximum volume of the safe which can be made by won.

.

and so

4. Two particles have locations at time on the -plane given by and . Find the minimum distance between and .

Therefore, .

Therefore, the minimum distance between the two particle is 0.

5. A closed cylindrical can is to hold of liquid. Find the height and radius that minimize the amount of material needed to manufacture the can.

6. Find a point on the curve that is closest to the point .

The distance from to an arbitrary point on the curve is

and the square of the distance is

.

.

Graphing on gives us a zero at , and so . The point on that is closest to is .

7. Find the largest area of a triangle which is inscribed in a circle of radius .

The area of the triangle is

Then,

Now

the maximum occurs where ,

so the triangle has height

and base .

8. Let be the volume of the right circular cone and be the volume of the right circular cylinder that can be inscribed in the cone. Find the ratio when the cylinder has the greatest volume.

, ( : radius )

By similar triangles, , so .

The volume of the cylinder is

.

Now .

So . The maximum clearly occurs when and then the volume is

9-10. Consider an ellipse .

9. Find the area of the rectangle of greatest area that can be inscribed in the ellipse.

Without loss of generality, choose

on the ellipse in first quadrant.

Thus, .

10. Find the minimum length of the tangent line which is cut by the axis and the axis.

,

Therefore, .

11. A cone-shaped paper cup is to be made to hold of water. Find the height and radius of the cup that minimizes the amount of paper needed to make the cup.

The volume and surface area of a cone with radius and height are given by and . We will minimize subject to

so

, so and hence has an absolute minimum at these values of and .

12. A pipe is being carried horizontally around a corner from a hallway wide into a hallway wide. What is the longest length that the pipe can have?

Let be the length of the line going from wall to wall touching the inner corner . As or , we have and there will be an angle that makes a minimum. A pipe of this length will just fit around the corner. From the diagram,

when

. Then and , so the longest pipe has length

.

13. Find the length of the shortest ladder that reaches over an 8 ft high fence to a wall which is 3 ft behind the fence.

,

when

when ,

when ,

so has an absolute minimum when , and the shortest ladder has length

, .