Question

Air is escaping from a balloon at the rate of $\mathrm{R}\left(\mathrm{t}\right)=\frac{60}{1+{\mathrm{t}}^2}{\mathrm{ft}}^3/\min$ , where $t$ is measured in minutes. How much air, in cubic feet, escapes during the first minute​?

Answer 1

Finding the volume of air escape:

Given air is escaping from a balloon at the rate of $\mathrm{R}\left(\mathrm{t}\right)=\frac{60}{1+{\mathrm{t}}^2}{\mathrm{ft}}^3/\min$

$$\begin{gathered} \therefore \mathrm{R}\left(\mathrm{t}\right)=\frac{60}{1+{\mathrm{t}}^2} \\ \Rightarrow \mathrm{R}\left(1\right)=\frac{60}{1+1^2}\mathbf{}\mathbf{[}\mathbf{\because }\mathbf{t}\mathbf{=}\mathbf{1}\mathbf{}\mathbf{min}\mathbf{}\mathbf{]}\mathbf{} \\ \Rightarrow \mathrm{R}\left(1\right)=\frac{60}{2} \\ =30{\mathrm{ft}}^3 \end{gathered}$$

Hence, $30{\mathrm{ft}}^3$ of air escapes during the first minute​.