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Chapter 8: Problem 17

The temperature, \(T\), of a chemical reaction is given by $$ T=120 \mathrm{e}^{0.02 t} \quad t \geq 0 $$ Calculate the time needed for the temperature to (a) double its initial value, (b) treble its initial value.

Short Answer

Expert verified
Answer: The time needed for the temperature to double its initial value is approximately 34.66 units, and the time needed to treble its initial value is approximately 54.72 units.

Step by step solution

01

Identify the initial temperature

The initial temperature \(T_0\) is given at \(t = 0\): $$ T_0 = 120 \mathrm{e}^{0.02(0)} = 120 $$
02

Find when temperature doubles

The temperature doubles its initial value, so \(T = 2 \cdot T_0 = 240\). Substitute this into the equation and solve for \(t\): $$ 240 = 120 \mathrm{e}^{0.02 t} $$ Now, divide both sides by 120: $$ 2 = \mathrm{e}^{0.02 t} $$ Take the natural logarithm of both sides to remove the exponential: $$ \ln(2) = \ln(\mathrm{e}^{0.02 t}) $$ Using the logarithmic identity, \(\ln(a^b) = b\ln(a)\), rewrite the right side: $$ \ln(2) = 0.02 t \ln(\mathrm{e}) $$ Since \(\ln(\mathrm{e}) = 1\): $$ \ln(2) = 0.02 t $$ Finally, divide both sides by 0.02 to find \(t\): $$ t = \frac{\ln(2)}{0.02} \approx 34.66 $$ So, the time needed for the temperature to double its initial value is approximately 34.66 units.
03

Find when temperature triples

The temperature triples its initial value, so \(T = 3 \cdot T_0 = 360\). Substitute this into the equation and solve for \(t\): $$ 360 = 120 \mathrm{e}^{0.02 t} $$ Now, divide both sides by 120: $$ 3 = \mathrm{e}^{0.02 t} $$ Take the natural logarithm of both sides to remove the exponential: $$ \ln(3) = \ln(\mathrm{e}^{0.02 t}) $$ Using the logarithmic identity, \(\ln(a^b) = b\ln(a)\), rewrite the right side: $$ \ln(3) = 0.02 t \ln(\mathrm{e}) $$ Since \(\ln(\mathrm{e}) = 1\): $$ \ln(3) = 0.02 t $$ Finally, divide both sides by 0.02 to find \(t\): $$ t = \frac{\ln(3)}{0.02} \approx 54.72 $$ So, the time needed for the temperature to treble its initial value is approximately 54.72 units.

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