3.3 Optimization Problems 7. A box with an open top is to be constructed from a rectangular piece of cardboard, 2m by 3m , by cutting out a square from each of the four corners and bending up the sides. Find the dimensions of the box corresponding to a maximum volume 8. Find the dimensions of the largest right-cylinder that can be inscribed in a cone of radius R = 6m and height H =9m .3.3 O llmlzatlon Problems 9. A north-south road intersects an east-west road at a point 0 . A motorcycle crosses 0 . at noon. travelling West at a constant speed of 80 krm'h . At the same time. a car is 501:: South of 0 . travelling North at 60 km!!! . Find the time at which they are closest to each other. and approximate the minimum distance between them. 10. The sum of two non-negative numbers is 16 . Find the maximum possible value and the minimum possible value of the sum of their cube roots. 3.4 Ogtlmlzatlon Problems In Economics and Science 3. The cost {in S) of manufacturing x thousands units of USB memory sticks is given by C(x) = 2x2 50x + 800 . How many items must be produced in order to minimize the unit cost u(x) = C(x) fx. 4. A farmer wishes to fence in a rectangular field of 60,000 ml. The North and the South fences cost $2M: while the East and the West fences cost $3! m . Find the dimensions of the field that will minimize the cost. 3.4 Optimization Problems in Economics and SCIENCE 5. A closed box with a square base is to contain 252cm3 . The bottom costs SiSi'cm2 . the top costs $23'cm2 . and the sides cost $3fc'm2 . Find the dimensions that will minimize the cost. 6. A soda cracker package {the top is closed) is to be constructed in the shape of a rectangular prism with a square base. The total capacity is 512 cm: . Find dimensions (length. width. and height} that will minimize the cost. 3.4 Optimization Problems In Economics and Science 1. A farmer wants to fence an area of 9600m2 in a rectangular field and divide it in half with a fence parallel to one of the sides of the rectangle. Find the dimensions (length and width} of the rectangular field that minimize the cost of the fence? 2. The selling price (in $} of an item is p(x) =600.02x , where x is the number of items sold per day. If the cost {in $} of manufacturing x items is 60:) = 1000+ 10x, nd the number of items to be manufactured per day in order to maximize the profit. 3.4 Oglmlzatlon Problems In Economics and Science 7. A cylindrical can {the top is closed} is to be made to hold 1000cm3 oi oil. Find the dimensions {radius and height) of the can that will minimize the cost of the metal to make the can. 8. Corn silos are usually in the shape oi a cylinder (with a closed base} surmounted by a hemisphere. If the volume of a silo is 1000m3 , what dimensions {radius and height) of the silo would use the minimum amount of materials? 3.4 Optimization Problems In Economics and Science 9. A tank has hemispherical ends and a Cylinder center. The cost (per square meter} of manufacturing the hemispherical ends is double in comparison with the cost {per square meter) of manufacturing the cylindrical part. Find the proportions {the ratio between height and radius} of the cylinder that will maximize the volume for a given total cost. 10. A lifeguard can run on the beach at Smls and swim at 4mm . If an incident happens at 40m from the shore and the lifeguard is on the shore at 100 m from the incident place, find the minimum amount of time it takes for the lifeguard to reach the place where the incident has occurred