A farmer is setting up separate pens for livestock by fencing off a rectangular area and running fence to create four parallel pens. If the farmer has a total of 500 yards of fence, what is the largest area that can be contained? Area = square yards A farmer is setting up separate pens for livestock by fencing off a rectangular area and running fence to create seven parallel pens. If the farmer has to enclose a total area of 1,600 square yards, what is the shortest length of fence needed? Total fence length = yards Open-box Problem. An open-box (top open) is made from a rectangular material of dimensions a = 14 inches by b = 11 inches by cutting a square of side x at each corner and turning up the sides (see the figure). Determine the value of x that results in a box the maximum volume. 12 10 987 A 2- + 7 8 b 10 11 12 13 14 15 Following the steps to solve the problem. Check Show Answer only after you have tried hard. (1) Express the volume V as a function of x: V = (2) Determine the domain of the function V of x (in interval form): (3) Expand the function V for easier differentiation: V = = (4) Find the derivative of the function V: V'= (5) Find the critical point(s) in the domain of V: (6) The value of V at the left endpoint is (7) The value of V at the right endpoint is (8) The maximum volume is V: = A piece of cardboard measuring 9 inches by 10 inches is formed into an open-top box by cutting squares with side length x from each corner and folding up the sides. Find a formula for the volume of the box in terms of x V(x) = Find the value for x that will maximize the volume of the box x = A box with a square base and open top must have a volume of 470596 cm. We wish to find the dimensions of the box that minimize the amount of material used. First, find a formula for the surface area of the box in terms of only x, the length of one side of the square base. [Hint: use the volume formula to express the height of the box in terms of x.] Simplify your formula as much as possible. A(x) = Next, find the derivative, A'(x). A'(x) Now, calculate when the derivative equals zero, that is, when A'(x) = 0. [Hint: multiply both sides by x .] A'(x) = 0 when x We next have to make sure that this value of x gives a minimum value for the surface area. Let's use the second derivative test. Find A"(x). A"(x) Evaluate A"(x) at the x-value you gave above. NOTE: Since your last answer is positive, this means that the graph of A(x) is concave up around that value, so the zero of A'(x) must indicate a local minimum for A(x). (Your boss is happy now.) If 2100 square centimeters of material is available to make a box with a square base and an open top, find the largest possible volume of the box. Volume: = cubic centimeters. A farmer was told by the municipality to enclose a field by a river to keep his animals out of the water. He decides to enclose a rectangular field as show below. The fence on the river side cost $12 per foot all other side cost $4 dollars per foot If he must enclose 202800 square feet, what values of x and y will maximize the enclosed area? x = feet y = feet River x X x y Q A fence 9 feet tall runs parallel to a tall building at a distance of 3 ft from the building as shown in the diagram. LADDER 9 ft 3 ft. We wish to find the length of the shortest ladder that will reach from the ground over the fence to the wall of the building. [A] First, find a formula for the length of the ladder in terms of 0. (Hint: split the ladder into 2 parts.) Type theta for 0. L(0) [B] Now, find the derivative, L'(0). Type theta for 0. L'(0) [C] Once you find the value of 0 that makes L'(0) = 0, substitute that into your original function to find the length of the shortest ladder. (Give your answer accurate to 5 decimal places.) L(0 min) ~ feet A rectangle is inscribed with its base on the x-axis and its upper corners on the parabola y = 10 - x. What are the dimensions of such a rectangle with the greatest possible area? Width = Height = A small house was built on an island off a perfectly straight shoreline. The point B on the shoreline that is closest to the island is exactly 5 miles from the island. Eight miles east of B is the closest source of fresh water to the island. A pipeline is to be built from the island to the source of fresh water by laying pipe underwater in a straight line from the island to a point Q on the shoreline between B and the water source, and then laying pipe on land along the shoreline from Q to the source. It costs 8000 dollars per mile to lay the pipe under the water and 5000 dollars per mile to lay pipe on land. How far east of B should Q be located in order to minimize the total construction costs? B underwater underground Find a formula for the cost of laying the pipe in terms of x C(x) = Find the value for x that will minimize the total cost. x = A Centerville is the headquarters of Greedy Cablevision Inc. The cable company is about to expand service to two nearby towns, Springfield and Shelbyville. There needs to be cable connecting Centerville to both towns. The idea is to save on the cost of cable by arranging the cable in a Y-shaped configuation. Centerville is located at (9,0) in the xy-plane, Springfield is at (0, 6), and Shelbyville is at (0, -6). The cable runs from Centerville to some point (x, 0) on the x-axis where it splits into two branches going to Springfield and Shelbyville. Find the location (x, 0) that will minimize the amount of cable between the 3 towns and compute the amount of cable needed. Justify your answer. To solve this problem we need to minimize the following function of x: f(x) = We find that f(x) has a critical number at x = To verify that f(x) has a minimum at this critical number we compute the second derivative "(x) and find that its value at the critical number is a positive number. , Thus the minimum length of cable needed is A cylinder shaped can needs to be constructed to hold 350 cubic centimeters of soup. The material for the sides of the can costs 0.02 cents per square centimeter. The material for the top and bottom of the can need to be thicker, and costs 0.06 cents per square centimeter. Find the dimensions for the can that will minimize production cost. Helpful information: h: height of can, r: radius of can Volume of a cylinder: V = rh Area of the sides: A = 2rh Area of the top/bottom: A = r To minimize the cost of the can: Radius of the can: Height of the can: Minimum cost: cents The lower edge of a 5 foot tall painting is 5 feet above your eye level (see picture below for the idea, but be aware it's not drawn to scale). painting 5 ft eye 5 ft At what distance should you stand from the wall so your viewing angle of the painting is maximized? ft Give your answer accurate to at least 2 decimal places The base of a rectangular box is twice as long as it is wide. width length The volume of the box is 108 cubic inches. The material for the top cost $0.08 per square inch, for the bottom it cost $0.04 per square inch, and for the sides cost $0.04 per square inch. Find the width, length, and height of the box to minimize the cost. Round answers to 2 decimal places. The width is inches The length is inches The height is inches A Norman window has the shape of a semicircle atop a rectangle so that the diameter of the semicircle is equal to the width of the rectangle. What is the area of the largest possible Norman window with a perimeter of 38 feet? ft You want to get from a point A on the straight shore of the beach to a buoy which is 65 meters out in the water from a point B on the shore. B is 70 meters from you down the shore. If you can swim at a speed of 5 meters per second and run at a speed of 7 meters per second, at what point along the shore, x meters from B, should you stop running and start swimming if you want to reach the buoy in the least time possible? Buoy B swim 70 m run Find a formula for the time spent in terms of x T(x) = Find the value for x that will minimize the total time spent running and swimming. x = A