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Warm-Up: March 14, 2013

Warm-Up: March 14, 2013. Solve the differential equation:. Homework Questions?. Exponential Growth and Decay. Section 6.4. Law of Exponential Change. If y changes at a rate proportional to the present amount ( dy / dt = ky ) and y(0)=y 0 , then The number k is called the rate constant

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Warm-Up: March 14, 2013

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  1. Warm-Up: March 14, 2013 • Solve the differential equation:

  2. Homework Questions?

  3. Exponential Growth and Decay Section 6.4

  4. Law of Exponential Change • If y changes at a rate proportional to the present amount (dy/dt = ky) and y(0)=y0, then • The number k is called the rate constant • Exponential growth if k>0 • Exponential decay if k<0

  5. Continuously Compounded Interest • where r is the continuous interest rate

  6. Example 1 • Find how long it would take for an investment to double at a continuous 4.9% interest rate.

  7. You-Try #1 • A couple wants to have an investment yield $500,000 when they retire in 12 years. How much do they need to invest if their expected continuous interest rate is 5.7% ?

  8. Radioactivity • Some elements or isotopes of elements are unstable and over time convert to a different element or isotope. • These elements/isotopes are called radioactive, and go through radioactive decay. • The amount of a radioactive substance is • The half-life is the time required for half of the sample to decay

  9. Half-Life • The half-life is the time required for half of a radioactive sample to decay.

  10. Example 2 • A clay pot was found to have 83% of its original Carbon-14. Estimate the age of the clay pot. The half-life of Carbon-14 is approximately 5730 years.

  11. You-Try #2 • The half-life of the radioactive element plutonium-239 is 25,000 years. A rock was found containing 1.8 grams of plutonium-239. Based on the mass and composition of the rock, a geologist concludes that it originally had 9.3 grams of plutonium-239. How old is the rock?

  12. Newton’s Law of Cooling • The rate at which an object cools (or warms) is proportional to the difference between its temperature (T) and the temperature of its surroundings (Ts).

  13. Example 3 • A hot cup of coffee is 54°C when it is poured. It is left sitting in a room with an air temperature of 22°C After 5 minutes, it has cooled to 42°C. How much longer will it take for the coffee to reach 37°C?

  14. You-Try #3 • A pan of warm water (46°C) was put into a refrigerator. Ten minutes later, the water’s temperature was 39°C; 10 minutes after that, it was 33°C. How cold was the refrigerator?

  15. Assignments • Read Section 6.4 (pages 330-337) • Page 338 Exercises #1-8 ALL, 13 • Page 339 Exercises #19-33 odd • Read Section 6.5 (pages 342-346)

  16. Warm-Up: March 15, 2013 • Prehistoric cave paintings were discovered in the Lascaux cave in France. The paint contained 15% of the original amount of Carbon-14. Estimate the age of the paintings at the time of the discovery. The half-life of Carbon-14 is approximately 5730 years.

  17. Homework Questions?

  18. Quiz: 6.1-6.3 • Clear everything off of your desk except pencil and eraser. • 15 minute time limit. • Any talking, looking at another student’s quiz, or using any electronic device will result in a zero.

  19. Newton’s Law of Cooling • The rate at which an object cools (or warms) is proportional to the difference between its temperature (T) and the temperature of its surroundings (Ts).

  20. Example 3 • A hot cup of coffee is 54°C when it is poured. It is left sitting in a room with an air temperature of 22°C After 5 minutes, it has cooled to 42°C. How much longer will it take for the coffee to reach 37°C?

  21. You-Try #3 • A pan of warm water (46°C) was put into a refrigerator. Ten minutes later, the water’s temperature was 39°C; 10 minutes after that, it was 33°C. How cold was the refrigerator?

  22. Resistance Proportional to Velocity • Some forces (such as air resistance) are proportional to the object’s velocity.

  23. Example 4 • Suppose an Iowa class battleship has a mass around 5.1x107 kg and a k value in Equation 2 of about 59,000 kg/s. Assume the ship loses power when it is moving at a speed of 9 m/s. • About how far will the ship coast before it is dead in the water?

  24. You-Try #4 • Suppose an Iowa class battleship has a mass around 5.1x107 kg and a k value in Equation 2 of about 59,000 kg/s. Assume the ship loses power when it is moving at a speed of 9 m/s. • About how long will it take the ship’s speed to drop to 1 m/s?

  25. Assignments • Read Section 6.4 (pages 330-337) • Page 338 Exercises #1-8 ALL, 13 • Page 339 Exercises #19-33 odd • Read Section 6.5 (pages 342-346)

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