A bacteria culture initially contains 100 cells and grows at a rate proportional to its size. After an hour the population has increased to 490. (a) Find an expression for the number of bacteria after t hours. P'(t} = (b) Find the number of bacteria after 2 hours. (Round your answer to the nearest whole number.) P(2) = |:] bacteria (c) Find the rate of growth after 2 hours. (Round your answer to the nearest whole number.) P'(2) = |:] bacteria per hour (d) When will the population reach 10,000? (Round your answer to one decimal place.) The half-life of cesium-137 is 30 years. Suppose we have a 100-mg sample. (a) Find the mass that remains after 1' years. y(t) = mg (b) How much of the sample remains after 90 years? (Round your answer to two decimal places.) :lmg (c) After how long will only 1 mg remain? (Round your answer to one decimal place.) A sample of a radioactive substance decayed to 90.5% of its original amount after a year. (Round your answers to two decimal places.) (a) What is the half-life of the substance? yr (b) How long would it take the sample to decay to 40% of its original amount? yrScientist can determine the age of ancient objects by a method called radiocarbon dating. The bombardment of the upper atmosphere by cosmic rays converts nitrogen to a radioactive isotope of carbon, 14C, with a half-life of about 5730 years. Vegetation absorbs carbon dioxide through the atmosphere and animal life assimilates 1'lC through food chains. When a plant or animal dies, it stops replacing its carbon and the amount of 14C begins to decrease through radioactive decay. Therefore, the level of radioactivity must also decay exponentially. A parchment fragment was discovered that had about 68% as much 14C radioactivity as does plant material on Earth today. Estimate the age of the parchment. (Round your answer to the nearest hundred years.) :3"