Moderate -0.3 This is a straightforward exponential growth/decay question testing standard techniques: finding the exponential constant from two data points, solving logarithmic equations, and differentiating exponentials. All parts are routine C3 procedures with no problem-solving insight required, making it slightly easier than average but not trivial due to the multi-step nature and calculator work needed.
The quantity \(X\) is increasing exponentially with respect to time \(t\). The table above shows values of \(X\) for different values of \(t\). Find the value of \(X\) when \(t = 20\). [3]
The quantity \(Y\) is decreasing exponentially with respect to time \(t\) where
$$Y = 80e^{-0.02t}.$$
Find the value of \(t\) for which \(Y = 20\), giving your answer correct to 2 significant figures. [3]
Find by differentiation the rate at which \(Y\) is decreasing when \(t = 30\), giving your answer correct to 2 significant figures. [3]
\begin{enumerate}[label=(\alph*)]
\item
\begin{tabular}{|c|c|c|}
\hline
$t$ & 0 & 10 & 20 \\
\hline
$X$ & 275 & 440 & \\
\hline
\end{tabular}
The quantity $X$ is increasing exponentially with respect to time $t$. The table above shows values of $X$ for different values of $t$. Find the value of $X$ when $t = 20$. [3]
\item The quantity $Y$ is decreasing exponentially with respect to time $t$ where
$$Y = 80e^{-0.02t}.$$
\begin{enumerate}[label=(\roman*)]
\item Find the value of $t$ for which $Y = 20$, giving your answer correct to 2 significant figures. [3]
\item Find by differentiation the rate at which $Y$ is decreasing when $t = 30$, giving your answer correct to 2 significant figures. [3]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{OCR C3 Q6 [9]}}