Hi can you please answer the following questions to help me study for my calculus test tomorrow? If possible with steps and explanation because I'm having a difficult time understanding what to do. I'll be sure to provide a good review!

Question 1:

A bacteria culture starts with 200 bacteria. After an hour the count has doubled to 400. Assume the population continues to increase at this rate. a) Write an equation that gives the number of bacteria, N, as a function of time, t, in hours. b) Determine the number of bacteria present after 5 hours. c) How fast is the number of bacteria increasing i) when they are initially discovered? ii) at the end of 5 hours? d) (i) How long will it take for the bacteria to reach a population of 10 000, to the nearest hundredth of an hour? (ii) How fast is bacteria population increasing at this point? e) At which point is the bacteria population increasing at a rate of 12 000 bacteria per hour? MCV4U 5. Laura has just bought a new motorcycle for $10 000. The value of the motorcycle depreciates t over time. The value can be modelled by the function V(t) =10000e7 , where Vis the value of the motorcycle after 1: years. a) At what rate is the value of the motorcycle depreciating the instant Laura drives it off the dealer's lot? b) Laura decides that she will stop insurance coverage for collision once the motorcycle has depreciated to one quarter of its initial value. When should Laura stop her collision insurance, to the nearest tenth of a year? c) At what rate is the motorcycle depreciating at the time determined in part b)? Example #1: A radioactive isotope of gold, flu-198, is used in the diagnosis and treatment of liver disease. Suppose that a 6.0 mg sample of Au-198 is injected into a liver, and that this sample decays to 4.6 mg after 1 day. Assume the amount of Au-198 remaining after t days is given by N(t) = Noe'\". a) Determine the disintegration constant for Au198, to 2 decimal places. Then write the equation for the amount of Au-198 remaining as a function of time. b) Determine the half-life of Au198, to the nearest tenth of a day. c) Write the equation that gives the amount of Au-I98 remaining as a function of time, in terms of its half-life. d) How fast is the sample decaying after 3 days, to the nearest tenth of a mg/day? MCV4U Example #2: A pendulum is an example of a harmonic oscillator, which is a moving object whose motion repeats over regular time intervals. When the amplitude of a harmonic osdllator diminishes over time due to friction, the motion is called damped harmonic motion. The vertical displacement of a sport utility vehicle's body after passing over a bump is modelled by the function h(t) = e'0-5'sint, where h is the vertical displacement, in metres, at time t, in seconds. (1) Determine when the maximum displacement of the SUV's body occurs, to the nearest tenth of second. b) Determine the maximum displacement, to the nearest hundredth of a metre. 11 4 200mg sample of a radioactive substance is placed into a nuclear reactor, After 2 weeks, the sample has decayed to 33% of its original amount. The amount of the radioactive substance remaining in the reactor can be modelled by the function N(t) = Noe , where t is the time in days. No = 200 N(+) = 6.33 t = 2weeks ( 14 days ) Q) Determine the value of the disintegration constant, 2, to four decimal places. Then write the equation for the amount of the substance remaining as a function of time. [4] b) Determine the half-life of the substance, to the nearest hundredth of a day. Then write the equation that gives the amount of the substance remaining as a function of time, in terms of its half-life. [4] A c) How fast is the sample decaying after 17 days, to the nearest tenth of a mg/day? [3]4. The concentration of a certain prescription drug in the bloodstream after a single dose can be modelled by the equation c = 100te'0'5', where c represents the concentration of the drug, in mg, and t represents the time in hours since the drug was taken. do a) Determine a . b) Determine the maximum concentration of the drug in the bloodstream, to the nearest tenth of o milligram, and the time when this occurs