The volume of a spherical hot air balloon expands as the air inside the balloon is heated. The radius of the balloon, in feet, is modeled by a twice-differentiable function $r$ of time $t$, where $t$ is measured in minutes. For $0 < t < 12$, the graph of $r$ is concave down. The table below gives selected values of the rate of change, $r^{\prime}(t)$, of the radius of the balloon over the time interval $0 \leq t \leq 12$.
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\begin{tabular}{ | c | | c | c | c | c | c | c | }
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\begin{tabular}{c} $t$ \\ (minutes) \end{tabular} & 0 & 2 & 5 & 7 & 11 & 12 \\
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\begin{tabular}{c} $r^{\prime}(t)$ \\ (feet per minute) \end{tabular} & 5.7 & 4.0 & 2.0 & 1.2 & 0.6 & 0.5 \\
\hline
\end{tabular}
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The radius of the balloon is 30 feet when $t = 5$.\\
(Note: The volume of a sphere of radius $r$ is given by $V = \frac{4}{3}\pi r^{3}$.)\\
(a) Estimate the radius of the balloon when $t = 5.4$ using the tangent line approximation at $t = 5$. Is your estimate greater than or less than the true value? Give a reason for your answer.\\
(b) Find the rate of change of the volume of the balloon with respect to time when $t = 5$. Indicate units of measure.\\
(c) Use a right Riemann sum with the five subintervals indicated by the data in the table to approximate $\int_{0}^{12} r^{\prime}(t)\, dt$. Using correct units, explain the meaning of $\int_{0}^{12} r^{\prime}(t)\, dt$ in terms of the radius of the balloon.\\
(d) Is your approximation in part (c) greater than or less than $\int_{0}^{12} r^{\prime}(t)\, dt$? Give a reason for your answer.