Calculus-Sec-4-4-Solution

4.4    Optimization Problems                         by SGLee - HSKim - JHLee

1. Find the 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 .

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 each  discount.

How much should they discount to maximize profits?

If  discount is made for each MP3 player sale,

.

The profit is maximized 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 using  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. Determine where the point  between  and  should be located to maximize the angle .

Let .

Then

.

Solving , we get . Since  we have .

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

Let r and t be as shown in the figure. The area of the triangle is

Then,

Now

The maximum area occurs when  with the height  and the base .

The largest area of a triangle is .

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.

Let  and  be the radius of the base  and height of the right circular cone let. Let  and  as show in the figure.  .

By similar triangles, , so .

The volume of the cylinder is

.

Now .

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

9. Consider an ellipse .

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.

Let = area of the rectangle and  = area of the ellipse.

Since ,

Thus, .

10. A closed cylindrical can is to be made to hold 100 of oil. What dimensions will use the least metal?

Let  be the radius and  the height (both in centimeters) of the can.

The total surface area  of the cylinder consisting of top, bottom and sides as a function with  and  is .

We eliminate the variable  by using the given condition .

Therefore   and .

The only critical number is .

Applying the First Derivative Test, we see that  has an absolute minimum at .

The corresponding value of  is   .

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 2 wide.

What is the longest length the pipe may 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,

where  and .

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 that is 3 ft behind the fence.

,

when

when ,

when ,

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

.            (*Label theta and L in the figure.)