A standard interior staircase has steps each with a rise (height) of 19 cm and a run (horizontal depth) of 23 cm. Research suggests that the stairs would be safer for descent if the run were, instead, 28 cm. For a particular staircase of total height 4.57 m, how much farther into the room would the staircase extend if this change in run were made?

The rise (height) of the step of the staircase,$h=19\mathrm{cm}$

The run (horizontal depth) of the step of the staircase,$d=23\mathrm{cm}$

The changed run of the step of the staircase =28cm

The total height of the staircase =4.57 m

To find the distance through which the staircase would extend further into the room, calculate the run for each step and multiply it by the number of runs of the staircase. Using the change in the run and the number of steps in the staircase, find the change in the run of the steps of the staircase.

The change in the run of the step of the staircase,

$$\begin{gathered} \nabla .\mathrm{d}=28\mathrm{cm}-23\mathrm{cm} \\ =5\mathrm{cm}\times \frac{1}{100\mathrm{cm}} \\ =0.05\mathrm{m} \end{gathered}$$

The total number of rises of the staircase is calculated as:

$$\begin{gathered} N=\frac{4.57m}{19cm\times {\displaystyle \frac{1m}{100cm}}} \\ =24 \end{gathered}$$

So, there are 24 runs of the staircase.

The staircase would extend through x is calculated as,

localid="1657703756726" $$\begin{gathered} \mathrm{x}=\mathrm{N}\times \nabla \mathrm{d} \\ =24\times 0.05\mathrm{m} \\ =1.2\mathrm{m} \end{gathered}$$

Thus, the staircase would extend through 1.2 m .