Optimization Review
1. Find two positive numbers whose product is 181 and whose sum is a minimum.
a.
b.
c.
d.
e.
2. Find two positive numbers such that the sum of the first and twice the second is 56 and whose product
is a maximum.
a.
b. 28 and 14
c.
d.
e.
3. Find the length and width of a rectangle that has perimeter
a.
b.
c.
d.
e.
meters and a maximum area.
12 m; 12 m.
16 m; 9 m.
1m; 23 m.
13 m; 11 m.
6 m; 18 m.
4. Find the length and width of a rectangle that has an area of 968 square feet and whose perimeter is a
minimum.
a.
b.
c.
d.
e.
5. Find the point on the graph of the function
all numerical values in your answer to four decimal places.
a.
b.
c.
d.
e.
that is closest to the point
. Round
6. Find the point on the graph of the function
that is closest to the point
.
a.
b.
c.
d.
e.
7. A rectangular page is to contain
square inches of print. The margins on each side are 1 inch. Find
the dimensions of the page such that the least amount of paper is used.
a.
b.
c.
d.
e.
8. Determine the dimensions of a rectangular solid (with a square base) with maximum volume if its
surface area is 529 square meters.
a.
Dimensions:
b.
Dimensions:
c.
Dimensions:
d.
Dimensions:
e.
Dimensions:
9. A Norman window is constructed by adjoining a semicircle to the top of an ordinary rectangular
window (see figure). Find the dimensions of a Norman window of maximum area if the total perimeter
is 38 feet.
a.
b.
c.
d.
e.
10. A rectangle is bounded by the x- and y-axes and the graph of
(see figure). What length and
width should the rectangle have so that its area is a maximum?
a.
b.
c.
d.
e.
11. A solid is formed by adjoining two hemispheres to the ends of a right circular cylinder. The total
volume of the solid is 23 cubic centimeters. Find the radius, r, of the cylinder that produces the
minimum surface area. Round your answer to two decimal places.
a.
b.
c.
d.
e.
12. The sum of the perimeters of an equilateral triangle and a square is 19. Find the dimensions of the
triangle and the square that produce a minimum total area.
a.
b.
c.
d.
e.
13. A sector with central angle is cut from a circle of radius 10 inches, and the edges of the sector are
brought together to form a cone. Find the magnitude of such that the volume of the cone is a
maximum.
a.
b.
c.
d.
e.
14. A farmer plans to fence a rectangular pasture adjacent to a river (see figure). The pasture must contain
720,000 square meters in order to provide enough grass for the herd. No fencing is needed along the
river. What dimensions will require the least amount of fencing?
a. x = 600 and y = 1200
b. x = 1000 and y = 720
c. x = 1200 and y = 600
d. x = 720 and y = 1000
e. none of the above
Answers
1. A
2. B
3. A
4. C
5. A
6. A
7. E
8. D
9. D
10. C
11. A
12. B
13. B
14. B