Concept explainers
Charging a Battery The rate at which a battery change is slower the closer the battery is to its maximum change
Where k is a positive constant that depends on the battery.
For a certain battery,
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College Algebra
- Radioactive Decay The half-life of a radioactive substance is the time H that it takes for half of the substance to change form through radioactive decay. This number does not depend on the amount with which you start. For example, carbon-14 is known to have a half-life of H=5770 years. Thus, if you begin with 1 gram of carbon-14, then 5770 years later you will have 12 gram of carbon-14. And if you begin with 30 grams of carbon-14, then after 5770 years there will be 15 grams left. In general, radioactive substances decay according to the formula A=A00.5tH Where H is the half-life, t is the elapsed time, A0 is the amount you start with the amount when t=0, and A is the amount left at time t. a. Uranium-228 has a half-life H of 9.3 minutes. Thus, the decay function for this isotope of uranium is A=A00.5t9.3, where t is measured in minutes. Suppose we start with grams of uranium-228. i. How much uranium-228 is left after 2 minutes? ii.How long will you have to wait until there are only 3 grams left? b. Uranium-235 is the isotope of uranium that can be used to make nuclear bombs. It has a half-life of 713 million years. Suppose we start with 5 grams of uranium-235. i. How much uranium-235 is left after 200 million years? ii. How long will you have to wait until there are only 3 grams left?arrow_forwardThe Beer-Lambert Law As sunlight passes through the waters of lakes and oceans, the light is absorbed, and the deeper it penetrates, the more its intensity diminishes. The light intensity I at depth x is given by the Beer-Lambert Law: I=I0ekx where I0 is the light intensity at the surface and k is a constant that depends on the murkiness of the water see page 402. A biologist uses a photometer to investigate light penetration in a northern lake, obtaining the data in the table. Light intensity decreases exponentially with depth. Use a graphing calculator to find an exponential function of the form given by the Beer-Lambert Law to model these data. What is the light intensity I0 at the surface on this day, and what is the murkiness constant k for this lake? Hint: If your calculator gives you a function of the form I=abx, convert this to the form you want using the identities bx=eln(bx)=exlnb. See Example 1b. Make a scatter plot of the data, and graph the function that you found in part a on your scatter plot. If the light intensity drops below 0.15 lumen lm, a certain species of algae cant survive because photosynthesis is impossible. Use your model from part a to determine the depth below which there is insufficient light to support this algae. Depth ft Light intensity lm Depth ft Light intensity lm 5 10 15 20 13.0 7.6 4.5 2.7 25 30 35 40 1.8 1.1 0.5 0.3arrow_forward
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