Calculus Writing Assignment: Optimization Each problem is worth 5 points. Total Points: 50 Answer each of the problems. Make sure to show all your work. 2 1. Find the formula that should be used to minimize the distance between y = 5x and the point (4, 3)? 2. What formula should be used to minimize the lateral surface area of a cone, where the volume 3 of the volume of the cone is 125 cm . V = r 2h Surface area of a cone is: S = r r 2 h2 . 3 3. The legs of a triangle are x and y. Find the equation that will maximize the area of the triangle given that 2x + y = 16? 4. Given y = x , find the closest point to 3 ,0 ? 2 5. Given the area of a rectangle is A = bh. If perimeter of the rectangle is 2b + 2h = 20, maximize the area of the rectangle. 2 6. Find the point on the parabola y = x that is closest to the point (0, 2). 7. Sue wants to build an enclosed area behind her house for her pets. One wall of the enclosed area will be the back of her house. She needs to the total to be 120 sq feet. She wants to minimize the cost of the fence materials. For the sides (W) fence materials cost $3 /ft, for the length (L) they cost $5 /ft. What dimensions should be fence be? 8. Find the largest possible area for a rectangle inscribed in a circle of radius 4. 9. A student wishes to maximize the amount of poster space for an art exhibit. The requirements are that the height and width must sum to 50. What should the dimensions of the poster be? 10. A piece of sheet metal is rectangular, 5 feet wide and 8 feet long. Congruent squares are to be cut from its four corners. The resulting piece of metal is to be folded and welded to form a box with an open top. How should this be done to get a box of largest possible volume