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Q1 A fisheries biologist is stocking fish in a lake

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Q1 A fisheries biologist is stocking fish in a lake. She knows that when there are n fish per unit of water, the average weight of each fish will be W (n) = 500 – 2n, measured in grams. What is the value of n that will maximize the total fish weight after one season? Hint: Total Weight = numbers of fish | average weight of a fish.

 

Q2.         A rectangle has a perimeter of 71 feet. What length and width should it have so that its area is a maximum? What is this maximum area?

 

Q3.         A designer of custom windows wishes to build a Normal Window with a total outside perimeter of 60 feet. How the window should be designed to maximize the area of the window. A Normal Window contains a rectangle bordered above by a semicircle.

 

Q4.         A ball thrown straight up in the air from ground level. Its height after t seconds is given by s (t) = -16 t² + 50t. When does the ball reach it maximum height? What is its maximum height?

 

Q5.         A farmer has 2,000 feet of fencing to enclose a pasture area. The field will be in the shape of a rectangle and will be placed against a river where there is no fencing needed. What is the largest are field that can be created and what are its dimensions?

 

Q6.         A telephone wire is to be laid from the telephone company to an island 7 miles off shore at a cost of $200,000 per mile along the shoreline and $300,000 per mile under the sea. How the wire should be laid at the least expensive cost if the distance along the shoreline is 12 miles. What is that cost?

 

Q7.         Alaina wants to get to the bus stop as quickly as possible. The bus stop is across a grassy park, 2,000 feet east and 600 feet north of her starting position. Alaina can walk along the edge of the park on the side walk at a speed of 6 feet/sec. She can also travel through the grass in the park, but only at a rate of 4 feet/sec (dogs are walked here, so she must move with care). What path will get her to the bus stop the fastest?

 

Q8.         I am 1 mile in the ocean and wish to get to a town 3 miles down the coast which is very rocky. I need to swim the shore and then walk along the shore. What point should I swim to along the shoreline so that the time it take to get to town is a minimum? I swim at 2 mph and walk at 4 mph.

 

Q9.         A nursery wants to add a 1,000-square-foot rectangular area to its greenhouse to sell seedlings. For aesthetic reasons, they have decided to border the area on three sides by cedar siding at a cost of $10 per foot. The remaining side is to be a wall with a brick mosaic that costs $25 per foot. What should the dimensions of the sides be so that the cost of the project will be minimized?

 

Q10.       An open box is to be made from a piece of metal 16 by 30 inches by cutting out squares of equal size from the corners and bending up the sides. What size square should be cut to create a box with greatest volume? What is the maximum volume as well?