1. An oil refinery is located on the north bank of a straight river that is2km wide. A pipeline is to be constructed from the refinery to storage tanks located on the south bank of the river7km east of the refinery. The cost of laying pipe is $400,000/km over land to a pointPon the north bank and $800,000/km under the river to the tanks. To minimize the cost of the pipeline, how far (in km) from the refinery shouldPbe located? (Round your answer to two decimal places.) km

2.A farmer wants to fence in a rectangular plot of land adjacent to the north wall of his barn. No fencing is needed along the barn, and the fencing along the west side of the plot is shared with a neighbor who will split the cost of that portion of the fence. If the fencing costs $18 per linear foot to install and the farmer is not willing to spend more than $9000, find the dimensions for the plot that would enclose the most area.

3.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, labeledx.)A window with the shape of a rectangle surmounted by a semicircle. The diameter of the semicircle is equal to the width of the rectangle. The rectangle has widthx.If the perimeter of the window is32feet,find the exact value ofx(in ft) so that the greatest possible amount of light is admitted.

4.The rate (in mg carbon/m³/h) at which photosynthesis takes place for a species of phytoplankton is modeled by the function

P =

120I

I² + I + 4

where I is the light intensity (measured in thousands of foot-candles). For what light intensity is P a maximum?I = thousand foot-candles

5.A box with a square base and open top must have a volume of 62,500 cm³. Find the dimensions of the box (in cm) that minimize the amount of material used.

Sides of base: cm

Height: cm

6.A rectangular storage container without a lid is to have a volume of 10 m³. The length of its base is twice the width. Material for the base costs $15 per square meter. Material for the sides costs $9 per square meter. Find the cost (in dollars) of materials for the least expensive such container. (Round your answer to the nearest cent.)

7.During the summer months Terry makes and sells necklaces on the beach. Last summer she sold the necklaces for $10 each, and her sales averaged 20 per day. When she increased the price by $1, she found that the average decreased by two sales per day.(a)Find the demand function (price p as (a) function of units sold x), assuming that it is linear.

p(x) =

(b)If the material for each necklace costs $4, what selling price should Terry set to maximize her profit?$ per necklac

8.A farmer wants to fence an area of 37.5 million square feet in a rectangular field and then divide it in half with a fence parallel to one of the sides of the rectangle. What should the lengths of the sides of the rectangular field be (in ft) in order to minimize the cost of the fence?

smaller value: ft

larger value: ft

9.A company operates 18 oil wells in a designated area. Each pump, on average, extracts 270 barrels of oil daily. The company can add more wells, but every added well reduces the average daily output of each of the wells by 9 barrels. How many wells should the company add in order to maximize daily production?

10.Find the point on the line

y=5x+2 that is closest to the origin.

(x,y) =

11.The top and bottom margins of a poster are each 12 cm and the side margins are each 8 cm. If the area of printed material on the poster is fixed at 1,536 cm², find the dimensions (in cm) of the poster with the smallest area.

width : cm

height: cm

If 30,000 cm² of material is available to make a box with a square base and an open top, find the largest possible volume (in cm³) of the box.

( please someone who knows what they are doing. Got them all wrong)