AP Calculus AB
More Optimization Problems
Name:_______________________________
Block: C or E 18 November 2016
Here are a few more mixed Optimization problems. Work on them with your group and then
finish them for homework.
1. A rectangular storage container with an open top is to have a volume of 10 m3. The length of
its base is twice the width. Material for the base costs $10 per square meter. Material for the
sides costs $6 per square meter.
a. Find the cost of materials for the cheapest such container.
b. Redo the calculations assuming the container has a lid that is made from the same
material as the sides.
2. a. Show that of all the rectangles with a given area, the one with the smallest perimeter is a
square.
b. Show that of all the rectangles with a given perimeter, the one with the greatest area is a
square.
3. Find an equation of the line through the point (3, 5) that cuts off the least area from the first
quadrant.
4. A steel pipe is being carried down a hallway 9 ft. wide. At the end of the hall
there is a right-angled turn into a narrower hallway 6 ft. wide. What is the length
of the longest pipe that can be carried horizontally around the corner? 5. Find the points on the ellipse 4x2 + y2 = 4 that are farthest away from the point (1, 0).
6. A Norman window has the shape of a rectangle surmounted by a semicircle. (Thus, the
diameter of the semicircle is equal to the width of the rectangle.) If the perimeter of the
window is 30 ft., find the dimensions of the window so that the greatest possible amount
of light is admitted. 7. A fence 8 ft. tall runs parallel to a tall building at a distance of 4 ft. from the building. What
is the length of the shortest ladder that will reach from the ground over the fence to the wall
of the building?
8. At which points on the curve y = 1 + 40x3 – 3x5 does the tangent line have the largest slope?
9. A rain gutter is to be constructed from a metal sheet of width 30 cm
by bending up one-third of the sheet on each side through an angle
θ. How should θ be chosen so that the gutter will carry the
maximum amount of water?
Some of these problems are from Single Variable Calculus with Vector Functions by James Stewart. Others are from old
handouts and their origins are lost in the mists of time.
More Optimization Problems
page 2
10. Where should the point P be chosen on the line segment AB so as to maximize
the angle θ? AB has a length of 3.
11. A manufacturer has been selling 1000 television sets a week at $450 each. A
market survey indicates that for each $40 rebate offered to the buyer, the number of sets
sold will increase by 100 per week.
a. Find the demand (price) function.
b. How large a rebate should the company offer the buyer in order to maximize its revenue?
c. If its weekly cost function is C(x) = 68,000 + 150x (where x is the number of $40 rebates),
how should the manufacturer set the size of the rebate in order to maximize its profit?
12. An artist is planning to sell signed prints of her latest work. If 50 prints are offered for sale,
she can charge $400 each. However, if she makes more than 50 prints, she must lower the
price of all the prints by $5 for each print in excess of the 50. How many prints should the
artist make in order to maximize her revenue?
13. In the planning of a sidewalk café, it is estimated that if there are 12 tables, the daily profit
will be $10 per table. Because of overcrowding, for each additional table the profit per table
(for every table in the café) will be reduced by $0.50. How many tables should be provided
to maximize the profit from the café?
14. A California distributor of sporting equipment expects to sell 10,000 cases of tennis balls
during the coming year at a steady rate. Yearly carrying costs (to be computed on the
average number of cases in stock during the year) are $10 per case, and the cost of placing
an order with the manufacturer is $80.
a. Find the inventory cost incurred if the distributor orders 500 cases at a time during the
year.
b. Determine the economic order quantity, that is, the order quantity that minimizes the
inventory cost.
15. Foggy Optics, Inc. makes laboratory microscopes. Setting up each production run costs
$2500. Insurance costs, based on the average number of microscopes in the warehouse,
amount to $20 per microscope per year. Storage costs, based on the maximum number of
microscopes in the warehouse, amount to $15 per microscope per year. Suppose that the
company expects to sell 1600 microscopes at a fairly uniform rate throughout the year.
Determine the number of production runs that will minimize the company's overall
expenses.
16. The marketing department of a business has determined that the demand for a product can
40
be modeled by p =
. The cost of producing x units is given by, C = 2 x + 50 . What
x
price will yield a maximum profit?
More Optimization Problems
page 3
ANSWER KEY (not checked!!!)
9
45
and Cost = $163.54
b. x = 3
and Cost = $191.28
2
16
! A$
2. a. A is a constant so P = 2l + 2 # & and the max/min occurs when l2 = A
"l%
" P − 2l %
b. P is a constant so A = l $
' and the max/min occurs when l = 14 P
# 2 &
1
3. y − 3 = ( x −1)
6
1. a. x = 3
4. 21.07045
" 1
4 2%
5. $ − , ±
' see diagram à
3 &
# 3
30
6. r =
≈ 4.2 and h is the same
4+π
7. 16.6477 ft
8. At (2, 225) and (–2, –223), the slope is 240
9.
π
3
10. 2.472 from point B
11. a. demand = 450 – 40x x = # of $40 rebates
b. Revenue = (450 – 40x)(1000 + 100x), max revenue at x = 5/8 (a rebate of $25)
c. Profit = Revenue – (68,000 + 150x), max profit at x = .60625 (a rebate of $24.25)
12. R(n) = (400 – 5n)(50 + n) where n = number of prints over 50, max at n = 15 (65 prints)
13. P(n) = (10 – 0.5n)(12 + n) where n = number of tables over 12, max at n = 4 (16 tables)
14. n = # of orders and c = # of cases per order so Inventory Costs = 80n + 10(c/2), also nc = 10,000
a. Inventory costs for 500 cases per order = $4100
b. best to have n = 25, c = 400
15. p = # of production runs, n = # made each run, pn = 1600
n
Cost = 2500 p + 20 ⋅ +15n = 2500 p + 25n so best to have p = 4
2
16. Profit = R − C =
40
⋅ x − (2x + 50) so best when x = 100, price is $4.
x