CEGEP CHAMPLAIN - ST. LAWRENCE
201-NYA-05: Differential Calculus
Patrice Camiré
Optimization
1. Find two positive real numbers whose sum is 24 and whose product is maximized.
Find two positive integers whose sum is 47 and whose product is maximized.
2. Find a positive number such that the sum of the number and its reciprocal is as small as possible.
3. Find two positive integers whose sum is 16 and so that the product of one number and the cube of the
other number is maximized.
4. What positive real number exceeds its forth power by the maximum amount?
5. A metal box is to have a square base and no top. It must be able to hold 500 cm3 . What should its
dimension be if the amount of metal used is to be as small as possible?
6. A company manufactures cardboard boxes. It has received an order for boxes having a square base
and a capacity of 1 cubic meter. If the cardboard used for the base, the sides and the top costs ¢5/m2 ,
¢3/m2 and ¢2/m2 respectively, what should the dimensions of the box be?
7. The college plans to build a track 400 m long around a rectangular field where both ends of the track
are semicircles. Find the dimension of the field so that its area is as large as possible.
8. A box with an open top is to be constructed from a square piece of cardboard 3 meters wide by cutting
a square from each of the four corners and bending up the sides. Find the largest volume such a box
can have.
9. A company wishes to build a cylindrical hot-water tank that can contain 20 cubic feet of water. What
are the dimensions of the tank that will maximize profit?
10. You wish to build a rectangular aquarium for which the depth is equal to the height. If you only
have 60 000 cm2 of glass and wish to maximize the amount of water the aquarium can contain, what
dimensions will yield the desired result?
11. Consider two straight edges of length a and b meeting at their endpoints and forming an angle θ.
Connect the other two endpoints to form a triangle. What choice of θ will yield a triangle with largest
area?
12. The strength of a rectangular beam is proportional to its width and the square of its height. What are
the dimensions of the cross section of the strongest beam that can be cut from a round log of radius
15 cm?
13. Find the point on the line y = 2x + 1 that is closest to the point (11, −2).
√
14. Find the point on the graph of the function y = x that is closest to the point (5, 0).
15. Find the point on the parabola y = x2 that is closest to the point (−1, 2).
16. Find the dimensions of the rectangle that can be inscribed in a semicircle of radius r > 0 with largest
possible area. (The base of the rectangle lies on the base of the semicircle.)
17. Find the dimensions of the rectangle of largest area that can be inscribed in an equilateral triangle of
side 8 meters if one side of the rectangle lies on the base of the triangle.
18. Show that a rectangle of maximum area for a fixed perimeter is always a square.
19. You are given a piece of rope 10 feet long. You wish to cut it into two pieces and with each piece,
form a circle. How should you cut the rope so as to minimize the total area of both circles? Does your
answer depend on the length of the rope you were given?
20. You are given a piece of rope 10 feet long. You wish to cut it into two pieces and form a square with
the first piece and form a circle with the second piece. How long should the first piece be so as to
minimize the total area of the square and circle?
21. Two posts, one 3 meters high and the other 12 meters high, stand 5 meters apart. They are to be
stayed by two wires, attached to a single stake, running from ground level to the top of each post.
Where should the stake be placed in order to use the shortest amount of wire?
22. A metal rain gutter is to have nine-centimeter sides and a nine-centimeter horizontal base, the sides
making equal angles with the base. How wide should the opening across the top be to maximize
carrying capacity?
23. You and your love are on opposite side of a river with very weak current (picture the river as being
horizontal). The river is 1 km wide and your love is tied to a tree 2 km east from the point directly
across from you on the other side of the river. Where should you reach the other side of the river so
as to rescue your love in the least amount of time? (Assume that you are swimming at 3 km/h and
walking at 5 km/h.)
24. Jane has a large garden in the summer that includes 100 blueberry plants. Over the years, she has
noticed that each plant produces 500 blueberries. Moreover, for each additional plant planted in the
garden, the production per plant drops by 4 blueberry. How many blueberry plants should Jane add
to her garden if she wants to maximize her production?
25. A company manufactures cylindrical aluminum cans that contain 200 cubic centimeters of soda pop.
The aluminum used for the top and bottom of a can is thicker and costs 5 cents per square centimeter,
while the aluminum used for the rest of a can costs 3 cents per square centimeter. What should the
dimension of a can be if the company wants to minimize its cost?
26. Consider the triangle formed by the following points in the xy-plane: (0, 1), (0, 3) and (x, 0) where
x > 0 is a variable. What value of x will maximize the angle corresponding to the vertex (x, 0)? What
is the value of the angle?
27. Determine the angle at which an arrow should be released from a bow so that the horizontal distance
travelled by the arrow is maximized. (Ignore air resistance.)
Answers
1. 12 and 12
23 and 24
2. 1
3. 4 and 12
√
4. 1/ 3 4
5. The sides of the base should be of length 10 cm and the height should be 5 cm.
6. The width of the box should be ≈ 0.95 m and its height ≈ 1.108 m.
7. The length of the field should be 100 m and the width of the field should be 200/π ≈ 63.66 m.
8. The largest volume is 2 m3 .
9. The radius should be r =
10 1/3
π
≈ 1.47 feet and the height should be
20
π
10 −2/3
π
≈ 2.94 feet.
10. The depth and height are equal to 100 cm and the length is equal to 133.3̄ cm.
11. The angle between both edges should be π/2, hence a right triangle will have the largest area.
√
√
12. The strongest beam should have width 10 3 cm and height 10 6 cm.
13. (1, 3)
√ x closest to the point (5, 0) is 29 , 3 2 2 .
√
√ 15. The point on the parabola y = x2 closest to the point (−1, 2) is − 1+2 3 , 1 + 23 .
14. The point on the graph of the function y =
√
√
2r and its height is r/ 2. The area of this rectangle is r2 .
√
17. The base of the rectangle should be 4 meters and the height 2 3 meters.
16. The length of the rectangle is
√
18. A = xy and P = 2x + 2y, so y = P/2 − x which implies that A = x(P/2 − x) = xP/2 − x2 . Then,
dA/dx = P/2 − 2x and so the only critical point is x = P/4. The scond derivative is −2 < 0 and so
x = P/4 is an absolute maximum. Solving for y yields y = P/2 − P/4 = P/4, thus x = y and the
rectangle is a square.
19. You should cut the rope in half.
No.
20. The length of the first piece should be 10/(1 + π/4) ≈ 5.6 feet.
21. The stake should be placed 1 meter away from the 3 m post.
22. The opening across the top should be 18 centimeters.
23. You should reach the other side of the river 1.25 km away from your love.
24. She can add either 12 or 13 blueberry plants.
25. The radius should be ≈ 2.67 cm and the height ≈ 8.91 cm.
√
26. x = 3 and θ = π/6
27. The angle should be π/4.