Name:_________________________________
Maus/Period:_________
Date:__________
Max/Min Problems 2
1 . Public health officials use rates of change to quantify the spread of an epidemic into an equation that
they then use to determine the most effective measures to counter it. A recent measles epidemic
followed the equation
, where y = the number of people infected and t = time in
days.
a) What is the domain of this function?
b) How many people are infected after 5 days?
c) What is the rate of spread after 5 days?
d) After how many days does the number of cases reach its maximum?
e) Use the above to sketch the graph of y.
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Fencing in Areas
2. Build a rectangular pen with three parallel partitions using 500 feet of fencing. What dimensions will
maximize the total area of the pen?
3. Farmer Al needs to fence in 800 sqyd, with one wall being made of stone which costs $24 per yard, and the
other three sides being wire mesh which costs $8 per yard. What dimensions will minimize cost?
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4. A family plans to fence in a rectangular patio area behind their house. They have 120 feet of fence to use.
One side of the rectangle is the back of the house. What should be the dimensions of the rectangular region
if they want to make the patio area enclosed as large as possible?
5. The owner of a warehouse decides to fence in an area of 800 square feet behind the warehouse. He plans to
use the wall of the building as one of the four sides that will enclose the rectangular area. He would like to
use the least amount of fencing necessary for the other three sides. How many feet of fence will be needed?
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6. A rancher has 200 feet of fencing to enclose two adjacent rectangular corrals. What dimensions should be
used so that the enclosed area will be maximum? What is the maximum area?
7. Construct a window in the shape of a semi-circle over a rectangle. If the distance around the outside of the
window is 12 feet, what dimensions will result in the rectangle having largest possible area?
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8. An open rectangular box with square base is to be made from 48 ft.2 of material. What dimensions will
result in a box with the largest possible volume?
9. An open rectangular box with square base is to be made from 48 ft.2 of material. What dimensions will
result in a box with the largest possible volume?
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1 0 . A mathematics book is to contain 3 6 square inches of oriented material per page with margins of
1 inch on each side and 1 .5 inches at the top and the bottom. Find the dimensions of the page that
will lead to the least amount of paper used per page.
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Boxes & Cans
11. A sheet of cardboard 3 ft. by 4 ft. will be made into a box by cutting equal-sized squares from each corner
and folding up the four edges. What will be the dimensions of the box with largest volume?
12. A cylindrical can is to hold 20 m.3 The material for the top and bottom costs $10/m.2 and material for the
side costs $8/m.2 Find the radius r and height h of the most economical can
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13. A candy box is to be made out of a piece of cardboard that measures 8 by 12 inches. Squares of equal size
will be cut out of each corner, and then the ends and sides will be folded up to form a rectangular box. What
size square should be cut from each corner to obtain a maximum volume?
14. A manufacturer of storage bins plans to produce some open-top rectangular boxes with square bases. The
volume of each box is to be 125 cubic feet. Material for the base costs $6 per square foot, and material for
the sides costs $3 per square foot. Determine the dimensions of the box that will minimize the cost of
materials.
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15. A manufacturer wants to design an open box that has a square base and a surface area of 108 square inches.
What dimensions will produce a box with maximum volume? What is the maximum volume?
16. A closed cylindrical container is to have a volume of 1000 cubic centimeters. It costs 5cents per square
centimeter for the top and bottom and 2 cents per square centimeter for the sides. Find the dimensions of
the container that costs the least.
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Business
17. The fuel, maintenance and labor costs (in dollars per mile) of operating a truck on an interstate highway are
described as a function of the truck’s velocity (miles per hour) by the algebraic rule:
.
What speed should the driver maintain on a 600 mile haul to minimize costs?
18. There are 50 apple trees in an orchard. Each tree produces 800 apples. For each additional tree planted in the
orchard, the output per tree drops by 10 apples. How many trees should be added to the existing orchard in
order to maximize the total output of trees?
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19. A car rental agency rents 200 cars per day at a rate of $30 per day. For each $1 increase in rate, 5 fewer cars
are rented. At what rate should the cars be rented to produce the maximum income? What is the maximum
income?
20. The total cost of producing x boxes of Badnasty candy bars is given by the function:
a) Find the marginal cost when the production level is 100 units .
b) What is the cost to actually produce the
unit.
c) Suppose these boxes of Badnasty bars sell for $10 each. Write the profit function, P(x), and use this to find
the maximum production level.
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Motion Problems
21. You are standing at the edge of a slow-moving river which is one mile wide and wish to return to your
campground on the opposite side of the river. You can swim at 2 mph and walk at 3 mph. You must first
swim across the river to any point on the opposite bank. From there walk to the campground, which is one
mile from the point directly across the river from where you start your swim.
What route will take the least amount of time?
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22. A hunter is at a point on a riverbank. He wants to get to his cabin, located 3 miles north and 8 miles east
of his current position. He can travel 5 mph on the river, but can only travel 2 mph on the rocky terrain.
How far up the river should he go in order to reach his cabin in the minimum time?
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23. Point B is on the edge of an island 6 miles due north of Point C on a straight shoreline. Point A is 9 miles
due east of Point C on the shore. A utility cable is to be run from Point A to Point B on the island. The cable
will go underground along the shoreline from Point A to a Point P between Points A and C. The cable will
then go under water from Point P to Point B. The cost to run the cable underground is $400 per mile while
the cost to run the cable under water is $500 per mile.
(a) How far from Point A should Point P be located in order to minimize the total cost of laying the cable?
(b) What is the minimum total cost?
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