ORTHOGONAL TRAJECTORIES
2. Compute the orthogonal trajectories of the family of curves given by
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where is an arbitrary constant.
Solution: Differentiating we get
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Elimination of
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At the point if any curve intersects orthogonally, then (if its slope is ) we must have
Solving this differential equation, we get
NEWTON’S LAW OF COOLING
2. At 9am , a thermometer reading 70ºF is taken outdoors, where the temperature is 15ºF. At 9:05am, the thermometer reading is 45ºF. At 9:10am, the thermometer is taken back indoors, where the temperature is fixed at 70ºF.
(a). Find the reading at 9:20am
(b). when the reading, to the nearest degree, will show the correct (70ºF) indoor temperature.
2a)
To = 70° F
Ta = 15° F
T(5) = 45° F
Determine k
45 - 15 = (70 - 15)e^-5k
30 = 55e^-5k
ln (30/55) / -5 = k
k = 0.1213
Find t(10)
T(10) - 15 = 55e^-1.213
T(10) = 55e^-1.213 + 15
T(10) = 31.4°
Apply new condtions
To = 31.4°
Ta = 70°
k = 0.1213
T(10) - 70 = (31.4 - 70)e^-
(0.1213)(10)
T(10) = -38.6e^-1.213 + 70
T(10) = 58.5°
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2b)
-1 = -38.6e^-(0.1213)t
t = ln ( 1/38.6 ) / -0.1213
t ≈ 30.1 minutes
GROWTH AND DECAY
EXAMPLE 4:
Suppose that an experimental population of fruit flies increases according to the law of exponential growth. There were 100 flies after day 2, and 300 after day 4. About how many flies were in the original population (at t = 0 days)?
EXAMPLE 5:
After stopping advertising, a company has its sales drop off. Initial sales are at 100,000 units, then after 4 months they fall to 80,000. Use the exponential decay model to predict 2 more months of decline.
SIMPLE INTEREST
A deposit is made to a bank account paying 8% interest compounded continuously. Payments are made from this account at a rate of $5000 per year. (a) Write a differential equation for the balance, B, in the account after t years (b) Write the solution to the differential equation. Use...