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Given an equation of the form $T=T_0 e^{kt}$ to model temperature, calculate the temperature after a given time, the time taken to reach a certain temperature, and the time taken for the temperature to double.

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The temperature $T$, in degrees $C$, of a chemical reaction is given by \\[ T = \\var{temp0}e^{\\var{k}t}, \\quad t\\geq0\\] where $t$ is time, measured in seconds.

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Part a:

To calculate the temperature at $\\var{time1}$ seconds, we use the formula given with $t=\\var{time1}$:

\\[ \\begin{split} T &\\,= \\var{temp0}\\,e^{\\var{k}\\times\\var{time1}} \\\\ &\\,= \\var{temp1}\\, ^\\circ C \\text{ (2 d.p.)} \\end{split} \\]

Part b:

To find how long it takes for the temperature to reach $\\var{temp2}\\, ^\\circ C$, we use the formula given with $T=\\var{temp2}$, and solve for $t$:

\\[ \\begin{split} \\var{temp0} \\, e^{\\var{k}t} &\\,= \\var{temp2} \\\\ e^{\\var{k}t} &\\,= \\simplify{{temp2/temp0}} \\\\ \\var{k} t &\\,= \\ln \\left(\\simplify[fractionNumbers]{{temp2/temp0}} \\right) \\\\ t &\\,=\\frac{\\ln \\left(\\simplify[fractionNumbers]{{temp2/temp0}}\\right)}{\\var{k}} \\\\ t &\\,= \\var{time2} \\text{ seconds (2 d.p.)}. \\end{split} \\]

Part c:

To calculate the time taken for the temperature to double from the initial temperature, we want to first check the initial temperature by setting $t=0$, which gives $T_0 = \\var{temp0} \\,^\\circ C$.

Therefore, we want to find the time taken for the temperature to reach $\\var{2*temp0} \\,^\\circ C$:

\\[ \\begin{split} \\var{temp0} \\, e^{\\var{k}t} &\\,= \\var{2*temp0} \\\\ e^{\\var{k}t} &\\,= 2 \\\\ \\var{k} t &\\,= \\ln (2) \\\\ t &\\,=\\frac{\\ln (2)}{\\var{k}} \\\\ t &\\,= \\var{timedub} \\text{ seconds (2 d.p.)}. \\end{split} \\]

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What is the temperature at $\\var{time1}$ seconds?

[[0]] $^\\circ \\,C$

(Give your answer to 2 decimal places where necessary)

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How long does it take for the temperature to reach $\\var{temp2} ^\\circ C$?

[[0]] seconds

(Give your answer to 2 decimal places where necessary)

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Calculate the time taken for the temperature to reach double the initial temperature.

[[0]] seconds.

(Give you answer to 2 decimal places where necessary)

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