Applications of the Derivative Advanced

show all work handwritten preferably.

1.

A rectangular garden will be fenced using 400 feet of fencing material. One side of the garden is partly covered by the barn and that part will not be fenced. 1f the barn is 50 feet long, what are the dimensions that will maximize the area of the garden? What is the largest possible area? Find the largest area for a rectangle that is inscribed in a semicircle with radius 4 inches. Find the largest possible area for a rectangle with base on thex axis, left side on the y axis, and upper right vertex on the curve y = 9 x 2. A closed top box with a square base is to have a volume of4000 cubic inches. What are the dimensions that will minimize the amount of: material used to build the box? A cylindrical can that has a capacity of 20 1113 will be made. The metal used to build the top costs $10 per square meter while the bottom costs $20 per square meter. The material used for the side costs $15 per square meter. What are the dimensions that will minimize the cost? 6. 3 adjacent playgrounds will be built using fencing; the areas of each will be equal. There is a total of 600 feet of fencing material, and the area is to be maximized. Let x and y represent the dimensions of each playground (the play grounds are adjacent sharing the side that has length y). Draw a diagram. Write the objective function in terms of X and y: Write the constraint in terms of X and y: 7. Let a and 1) two positive numbers with 8a + 7b = 60 . Find the maximum value for their product, ab. 8. CBC is a right triangle with right angle C. This triangle's base is on the positive x-axis; the vertex 0 is at the origin, and the vertex C is on the positive x-axis. The vertex B is on the line 31 + 5x = 10. Find the maximum area of this triangle. Draw a diagram first. 9. OBCD is a rectangle with corner 0 at the origin, corner B on the positive y-axis, corner D on the positive xaxis. The corner C is on the line 2y + 3x = 12. Find the maximum area of this rectangle. Draw a diagram first. 10. A box with a square base and closed top has a fixed surface area of 120 cm2. Find the dimensions of the box that give the maximum volume.11. An open box to be formed from a piece of metal 16 by 30 inches by cutting out squares of equal sides from the corners and bending up the sides. What size square should be cut out to create a box of greatest volume? What is the maximum volume? Given f'(x) = (x2 +7)' 1/3 and f (1) =2, use differentials to estimate f (0.8).Use differentials to approximate the change in the volume of a spherical balloon of radius 2 meters if the balloon deflates to a radius of 1.8 meters