91Ó°ÊÓ

The vertical height is $h=16m$.

Mass of Tatiana is $m_T=50.2kg$.

Mass of Bill is $m_B=68.2\mathrm{kg}$.

Mass of Ricardo is $m_R=81.8\mathrm{kg}$.

Mass of Melanie is $m_M=59.1\mathrm{kg}$.

From the diagram,

Time taken by Tatiana is $T_T=31\mathrm{s}$.

Time taken by Bill is $T_B=50\mathrm{s}$.

Time taken by Ricardo is $T_R=45\mathrm{s}$.

Time taken by Melanie is $T_M=53\mathrm{s}$.

The average amount of work done or energy converted per unit time is known as average power output that is given by,

${\mathit{P}}_{\mathbf{a}\mathbf{v}\mathbf{g}}\mathbf{=}\frac{\mathbf{W}}{\mathbf{T}}$ ...........(1)

Here, W is work done, and T is time.

The work done against the gravity to reach height h is given by,

$\mathit{W}\mathbf{=}\mathit{m}\mathit{g}\mathit{h}$ ..........(2)

Here, m is mass, and g is acceleration due to gravity.

From equations (1) and (2),

$P_{\mathrm{avg}}=\frac{mgh}{\mathrm{T}}$ ..........(3)

Therefore, the required time is 3.64 hr.

Find the average power output for Tatiana as follows.

$$\begin{gathered} P_{{\mathrm{avg}}_{\mathrm{T}}}=\frac{m_{T}gh}{{\mathrm{T}}_{\mathrm{T}}} \\ =\frac{\left(50.2kg\right)\left(9.8m/s^2\right)\left(16m\right)}{31s} \\ =253.9\mathrm{watt} \end{gathered}$$

Find the average power output for Bill as follows.

$$\begin{gathered} P_{{\mathrm{avg}}_B}=\frac{m_{B}gh}{{\mathrm{T}}_B} \\ =\frac{\left(68.2kg\right)\left(9.8m/s^2\right)\left(16m\right)}{50s} \\ =213.89\mathrm{watt} \end{gathered}$$

Find the average power output for Ricardo as follows.

$$\begin{gathered} P_{{\mathrm{avg}}_R}=\frac{m_{R}gh}{{\mathrm{T}}_R} \\ =\frac{\left(81.8kg\right)\left(9.8m/s^2\right)\left(16m\right)}{45s} \\ =285\mathrm{watt} \end{gathered}$$

Find the average power output for Melanie as follows.

$$\begin{gathered} P_{{\mathrm{avg}}_M}=\frac{m_{M}gh}{{\mathrm{T}}_M} \\ =\frac{\left(59.1kg\right)\left(9.8m/s^2\right)\left(16m\right)}{53s} \\ =175.85\mathrm{watt} \end{gathered}$$

From the above calculations, Ricardo has the largest average power output of $P_{{\mathrm{avg}}_R}=285\mathrm{watt}$, and Melanie has the smallest average power output of $P_{{\mathrm{avg}}_M}=175.85\mathrm{watt}$.

Therefore, Ricardo has the largest average power output, and Melanie has the smallest average power output

It is given that Chang is very fit, and has a mass of $m_C=62.3\mathrm{kg}$and his average power output is$P_{av{g}_C}=1\mathrm{hp}$ or $P_{{\mathrm{avg}}_C}=746\mathrm{watt}$.

From equation (3),

$$\begin{gathered} T_C=\frac{m_{C}gh}{P_{av{g}_C}} \\ =\frac{\left(62.3\mathrm{kg}\right)\left(9.8\mathrm{m}/{\mathrm{s}}^2\right)\left(16\mathrm{m}\right)}{746\mathrm{watt}} \\ =13.1s \end{gathered}$$

Therefore, it will take 13.1 s to run up the stairs.