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.
http://matrix.skku.ac.kr/cal-lab/cal-4-4-exs-5.html
.
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.)
