### Episode 6: Solutions

a) dT/dh = k(T-5050)

Separate the variables.

dT(1/(T-5050)) = k dh

Antidifferentiate both sides of the equation.

ln(T-5050) = kh +C

T -5050 = e^(kh+C)

b) Using the general solution, and the points (0,5000) and (150,3610.5405), solve for the exact equation.

T = e^(kh+C) + 5050

First use the point (0,5000) to solve for one of the variables.

5000 = (e^(k(0)))(e^(C)) +5050

-50 = e^(C)

T = -50e^(kh) + 5050

Now using the other point (150,3610.5405), solve for the last unknown.

3610.5405 = -50(k(150)) +5050

-1439.4595 = -50e^(150k)

Solve for k.

-1439.4595/-50 = e^(150k)

28.7892 = e^(150k)

ln(28.7892) = 150k

k = (1/150)ln(28.7892)

k = 0.0224

c) Using the fact that the element can only be used at 4444 Kelvin, you have to insert that into the equation and solve for the height,

T = -50e^(0.0224h) +5050

4444 = -50e^(0.0224h) +5050

-606 = -50e^(0.0224h)

-606/-50 = e^(0.0224h)

12.12 = e^(0.0224h)

ln(12.12) = 0.0224h

h = (1/0.0224)ln(12.12)

h = 111.3775 cm

Separate the variables.

dT(1/(T-5050)) = k dh

Antidifferentiate both sides of the equation.

ln(T-5050) = kh +C

T -5050 = e^(kh+C)

*T = e^(kh+C) + 5050*b) Using the general solution, and the points (0,5000) and (150,3610.5405), solve for the exact equation.

T = e^(kh+C) + 5050

First use the point (0,5000) to solve for one of the variables.

5000 = (e^(k(0)))(e^(C)) +5050

-50 = e^(C)

T = -50e^(kh) + 5050

Now using the other point (150,3610.5405), solve for the last unknown.

3610.5405 = -50(k(150)) +5050

-1439.4595 = -50e^(150k)

Solve for k.

-1439.4595/-50 = e^(150k)

28.7892 = e^(150k)

ln(28.7892) = 150k

k = (1/150)ln(28.7892)

k = 0.0224

**T = -50e^(0.0224h) +5050**c) Using the fact that the element can only be used at 4444 Kelvin, you have to insert that into the equation and solve for the height,

*.***h**T = -50e^(0.0224h) +5050

4444 = -50e^(0.0224h) +5050

-606 = -50e^(0.0224h)

-606/-50 = e^(0.0224h)

12.12 = e^(0.0224h)

ln(12.12) = 0.0224h

h = (1/0.0224)ln(12.12)

h = 111.3775 cm

**The height that the element has to be held at in order to reforge the sword is 111.3775 cm.**
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