Exponential Models with Differential Equations Answer the following 10 questions. Be sure to show all of your work and provide proper calculus notation. You must tie your answers back to the problem using full sentences and include proper units. 1. The radioactive decay of an isotope can be modeled by the following differential equation, where t is measured in years. Find the half-life of the isotope. dy dt = -0.039y 2. A particular isotope has a half-life of 35 minutes. If the decay of the isotope is modeled by the following differential equation, where t is measured in minutes, what is the decay constant value of k? dy dt = - - ky 3. Suppose that a bacteria colony grows so that the population increases exponentially with time. At the end of 5 hours, there are 12,000 bacteria. At the end of 10 hours, there are 50,000 bacteria. How many bacteria were present initially? 5. The temperature of an ingot of gold is 70C above room temperature right now. Twenty minutes ago, it was 80C above room temperature. How far above room temperature will the gold be: a. 15 minutes from now? b. 2 hours from now? c. When will the gold be 10C above room temperature? 6. A experimental insect population increases according to the law of exponential growth. There were 200 insects after the third day of the experiment and 500 insects after the fifth day. Approximately how many insects were in the original populations?