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Practical Exponentials

Exponential decay is a real phenomenon with many practical applications – radioactive decay, Newton's Law of Cooling, measuring and controlling the thickness of metals – since the intensity of radiation decays exponentially with penetration into the metal.. Radioactivity is used in smoke alarms to – too low a level of radiation detected will set off the smoke alarm. Because the alarm must have a certain lifespan, the activity must be calculated at this point and the detector calibrated accordingly. The general equation for exponential decay iswhere

is the background temperature or background level of radioactivity for example.

is the initial excess temperature or level of radioactivity for example.

is the decay constant.

Example: The temperature of a cup of coffee is give at any time t minutes by the equation

a)Write down the room temperature.

b)Find the temperature when

c)Find the rate of change of temperature whenInterpret the sign.

d)Find the time when the coffee has cooled halfway.

a)The room temperature is the temperature when the coffee has cooled completely ie

b)

c)We must differentiate to findThe constant term, 20, disappears andWe now substitute the value of t into this equation.The minus sign means the temperature is falling.

d)When the coffee has cooled halfway, the excess temperature has fallen by half – it is nowThe temperature isWe solve