## Related questions with answers

Direct measurements of the atmospheric carbon dioxide $\left(\mathrm{CO}_2\right)$ concentration have been made since about $1959$, when the measured concentration was about $316$ parts per million. In $2017$, the concentration was about $407$ parts per million. Assume that this growth can be modeled with an exponential function of the form $Q=Q_0 \times(1+r)^t$. a. By experimenting with various values of the fractional growth rate $r$, find an exponential function that fits the given data for $1959$ and $2017$. b. Use this exponential model to predict when the $\mathrm{CO}_2$ concentration will be $560$ parts per million (twice its preindustrial level). c. Research recent trends in the carbon dioxide concentration. Does your model seem to fit, overestimate, or underestimate its recent rise? Explain.

Solution

Verified$(a)$ As we are observing the change of concentration of carbon dioxide of $91$ part per million over the course of $58$ years, we are dealing with a relatively small growth rate. We can use a trial and error method on an equation

$C=316\left(1+r\right)^{58}$

to determine the appropriate growth rate to get the concentration of $C=407$

$\begin{align*} r&=1\%=0.01&\longrightarrow C&=562.76\\ r&=0.5\%=0.005&\longrightarrow C&=422\\ r&=0.4\%=0.004&\longrightarrow C&=398.33\\ r&=0.45\%=0.0045&\longrightarrow C&=410\\ r&=0.44\%=0.0044&\longrightarrow C&=407.64\\ r&=0.43\%=0.0043&\longrightarrow C&=405.29\\ r&=0.437\%=0.00437&\longrightarrow C&=406.9\\ \end{align*}$

So, we can conclude that the approximate growth rate is $r=0.437\%$.

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