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\[P.E.=mgh\]

Where m is the mass of the object, g is the acceleration due to gravity, and h is the height of the object from the ground.

Power of an object is the rate of energy per unit time.

The power P of an object is given by:

\[P=\dfrac{mgh}{t}\]

Where m is the mass of the object, g is the acceleration due to gravity, h is the height of the object from the ground, and t is the time in seconds.

Mass of the boy, \[m=50\text{ kg}\]

Time taken to run up the stairs, \[t=9\text{ s}\]

The staircase has \[45\] steps each of height \[15\text{ cm}\].

So, the height of the stairs, \[h=45\times 15\text{ cm}=675\text{ cm}=6.75\text{ m}\]

\[g=10\text{ m/}{{\text{s}}^{2}}\]

Substituting the value of mass, acceleration due to gravity, height and time in the power- formula:

\[\begin{align}

& P=\dfrac{(50\text{ kg)(10 m/}{{\text{s}}^{2}})(6.75\text{ m)}}{9\text{ s}} \\

& P=\dfrac{3375\text{ J}}{9\text{ s}} \\

& P=375\text{ W} \\

\end{align}\]

The potential energy of the boy at the base of the staircase is zero as his height from the ground is zero.

Work done, \[W=F\cdot S\]

Where the force F on the body is the weight of the boy.

So, \[F=mg\]

The displacement S of the boy is equal to the height of the staircase.

Power of the boy is the rate of work done.

So, \[P=\dfrac{W}{t}=\dfrac{(mg)(h)}{t}=\dfrac{mgh}{t}\].