Please answer the following:

('1 point} Suppose f{$) = 1.1113. (a) Find the linear approximation of f at a: = 1. L($) = (b) Use the linear approximation to estimate In 1.13. 1111.13 m ('1 point} Suppose at) 2 . (a) Find the equation of the tangent line (Le. the linear approximation) to f at a = 36. y = 97+ (b) Rounding to 4 decimals, use the result in part (a) to approximate: vii/35.1 = ('1 point}A person is pulling a boat into clock with a rope. If the person is pulling from a height of 1 m above the boat and the rope is being pulled at a rate of 1.2 mis, how fast is the boat approaching the clock when it is B m from the dock? Answer: m/s Note: Express your answer as a positive speed. ('1 point} A spherical snowball is melting so that its diameter is decreasing at rate of 0.3 omfmin. At what rate is the volume of the snowball changing when the diameter is '14 cm. Answer: (:m3/m1'n [Hint: Volume of a sphere with radius r is V 2 gm} .] ['1 point} Gravel is being dumped from a conveyor belt at a rate of 50 m3g'mz'n. It forms a pile in the shape of a right circular cone whose diameter and height are always equal. How fast is the height of the pile increasing when the pile is 14 m high? Answer: mjmin [Hint: The volume of a cone with height h and radius r is given by V = 1rr2hi ('1 point}A street light is at the top of a 11 ft tall pole. A 6ft tall woman walks away from the pole (along a straight path) at a speed of 5 ftisec. How fast is the tip of her shadow moving when she is 45 ft from the base of the pole? Answer: ftfsec dN ('1 point} Suppose N(t) denotes a population size at time t where the E = 0.02N(t). If the population size at time t = 4 is equal to 100, use a linear approximation to estimate the size of the population at time t = 4.1. L(4.1) = ('1 point} Suppose t) is the weight (in grams) of a solid sitting in a beaker of water and the solid is dissolving at a rate (in gramsfminute) of f'(t) = 1f(t}(4 + t\". If there is 5 grams of solid at time t = 2 minutes, use a linear approximation to estimate the solid's weight 1 second later. Answer: grams ('1 point} The half-life of Radium-226 is 1590 years. If a sample contains 300 mg. how many mg will remain after 2000 years? Answer: mg ['1 point} Human hair from a grave in Africa proved to have only 51% of the carbon 14 of living tissue. If the half-life of carbon 14 is 5730 years, determine how long ago the body was buried. Answer: years ago. ('1 point} A bacteria culture starts with 800 bacteria and grows at a rate proportional to its size P[t) . After 3 hours, there are 6000 bacteria. (a) Find an expression for the number of bacteria after t hours. P(4) = bacteria (c) Find the growth rate after 4 hours. P'[4) = bacteriafhour (d) After how many hours will the population reach 30000? t 2 hours ('1 point} Determine the value of $200 in 5 years if it's invested at 4% compounded: (a) Quarterly Answer: $ (b) Continuously Answer: $ [1 point}Acup of ooffee at 173 degrees is poured into a mug and left in a room at 72 degrees. After minutes, the ooffee is 140 degrees. Assuming that Newton's Law of Cooling applies: (a) Determine the temperature of the coffee after 16 minutes. T(l) = degrees (b) After how many minutes will the ooffee be '100 degrees? t 2 minutes