HW 4 –6 Optimization
Math 400!
Name____________________
1. Find two positive numbers whose sum is 9 if the product of one number and the square of the
other number is the maximum possible product.
2 . A farmer wants to fence a rectangular piece of land. One side of the rectangle plot will not have
any fence on it as it fronts a river with a straight bank. The other 3 sides will use 500 feet of
fencing to enclose the largest area possible. Find the length and width of the rectangular plot and
the maximum area it encloses.
y
x
!
HW 4 – 6!
river
no fence
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3. A farmer wants to make a rectangular pen that encloses a total area of 216 square feet . He wants
2 pens of equal size inside the rectangular pen as shown. Find the dimensions of the outside
rectangular pen and the amount of fencing needed.
y
y
x
x
x
y
y
4. A farmer wants to make a rectangular pen using a total of 500 feet of fencing. He wants to have
three parallel partitions inside the pen as shown below. Find the dimensions of the outside
rectangular pen that will maximize the total area of the pen ?
y
x
y
x
y
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y
x
y
y
x
y
x
y
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5. A piece of cardboard is 14 inches by 10 inches. A square is cut out of each corner as shown
below. The sides of the remaining shape are then folded up to form a box. Determine the height
of the box that will give a maximum volume.
12
x
x
x
x
x
x
x
x
14
6. You want to make the box shown below with an open top and a square base that contains the
maximum volume using 48 square cm of material for the sides and bottom . What are the
dimensions of the box and what is the largest volume the box can contain?
HW 4 – 6!
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7. You want to make a tank with a rectangular bottom and rectangular sides that will be open at the
top. It is constructed so its width is 4 meters and its volume is 36 cubic meters. The materials to
build the tank cost $10 per square meter for the bottom and $ 5 per square meter for the sides.
Find the cost of the least expensive tank?
8. A box is made from two different types of material. The cheaper material is used for the sides of
the box and costs $ 6 a square foot. The more expensive material is used for the top and bottom
of the box and costs $10 a square foot. Additionally, the length of the box will be twice its width
and the box must hold a volume of 40 cubic feet. Find the dimensions which will minimize the cost
of the box and what the minimum cost will be.
HW 4 – 6!
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9. A container in the shape of a right circular cylinder with no top has an outside surface area of 3π
square feet. What height h and base radius r will maximize the volume of the cylinder ?
10. A cylindrical can is to hold 20 π cubic meters. The material for the top and bottom costs $10 a
square meter and material for the side costs $ 8 a square meter. Find the radius r and height h of
the can with the minimum cost.
HW 4 – 6!
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11. A Florida citrus grower has 60 orange trees planted and his average yield per tree is 400
oranges for a yield of 24000 oranges. For each new tree he planes the average yield will
decrease by 4 oranges per tree. How many new trees should he plants to maximize the total
yield.
Yield = number of tree * yield per tree.
let n be the number of new trees added.
12. The city of New York charges $10 for a one day ticket to park in a parking space. It is
averaging 27,000 ticket sales a day. A recent research paper concluded that for each 10 cents
decrease in price the number of new tickets sold would increase by 300. What should the ticket
price be sat at to maximize revenue?
Revenue = price * number of tickets sold.
Let n = the number of times the price is reduces by 10 cents.
HW 4 – 6!
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13. Find the point on the parabola y =
D=
1 2
x that is closest to the point ( 1 , 4 )
2
( x − x1)2 + ( y − y1)2
14. A landscape architect plans to enclose a 3000 square foot rectangular region in a botanical
garden, She will use shrubs costing $ 15 per foot along three sides and fencing costing $10 per
foot along the fourth side. Find the minimum total cost.
HW 4 – 6!
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