Calculus Ab: Answer the following questions with each steps and with number line. what kind of derivative is used ( product rule, chain file,etc) Some questions are optimization.

8. An open-top rectangular box is constructed from a 10 in by 16 in piece of cardboard by cutting up squares of equal side length from the corners and folding up the sides. Find analytically the dimensions of the box of largest volume and the maximum volume. 16. A five-foot ladder is falling down a wall. The height of the ladder is decreasing by - ft/sec. When the top of the ladder is 4 feet from the ground, how fast is the base of the ladder moving away from the house? 9. Find the surface area of the smallest open-top storage bin with a square base and vertical sides that can be made with a volume of 108 fts of sheet steel. 17. A 6-foot-tall man walks away from a 20-foot lamp post at the rate of 5 ft/sec. The light at the top of the post casts a shadow in front of the man. How fast is the length of his shadow changing while he is walking? 10. A rectangle is inscribed under one of arch of y = 8 cos(0.3x) with the base on the x-axis and its upper two vertices on the curve above the x-axis. What is the largest area the rectangle can have ? 11. lim 18. Find if the Mean Value Theorem can be used over the interval [-4,4]. If it can, find the value c 12. lim x-0 x tan x which satisfies the theorem. a) f ( x) = x +1, x52 (4x2 - 7, x > 2 im sin(5x) b) g(x) = tan(nx) + 4 14. Let y = ex. a ) Find dy c) h(x) = x3+ 3x2 - 4x+ 4 b) Estimate dy for x = 2 and dx = 0.04. 15. The base of a cone shaped tank is a circle with a diameter of 12 feet, and the height of the cone et. The tank is filled to a depth of 8 feet, and water is flowing out of the tank at the rate of 5 cubic feet per minute. Find the rate of change of the depth of water in the tank