Chapter 9.2, Problem 53PS

Solution

To find: the length of the arc formed by the outer edge of a step.

The length of arc is $16.49\text{ inches}\text{.}$

Given:

It is given that s spiral staircase has $15$ steps. Each step is a sector with a radius of $42\text{ inches}$ and a central angle of $\frac{\pi }{8}.$

Formula Given:

  $\text{Arc length}\left(s\right)=r\theta$

where,

  $\begin{array}{l}r=\text{radius}\\ \theta =\text{central angle }\end{array}$

Calculation:

Calculating the length of the arc as:

  $\begin{array}{l}s=r\theta \\ s=42\left(\frac{\pi }{8}\right)\\ s=42\times \frac{3.14}{8}\\ s=\frac{131.88}{8}\\ s=16.49\end{array}$

Hence, the length of arc is $16.49\text{ inches}\text{.}$

To find: that through what angle would rotate by climbing the stairs.

  ${360}^{\circ }$

Given:

It is given that spiral staircase has $15$ steps. Each step is a sector with a radius of $42\text{ inches}$ and a central angle of $\frac{\pi }{8}.$

Calculation:

It is given that there are $15$ steps, and each step in the spiral staircase has a central angle of $\frac{\pi }{8}.$ Also, there is a sixteenth turn for stepping up on the landing.

Hence, the total angle rotation by the staircase and landing would then be equal to $16\left(\frac{\pi }{8}\right)\text{ or }2\pi ,$ which is equivalent to ${360}^{\circ }$ .

So, a person climbing the spiral staircase would rotate at an angle of ${360}^{\circ }$ .

To find: the square inches of carpeting would need to cover the $15$ steps.

  $5192.775{\text{ in}}^2$

Given:

It is given that s spiral staircase has $15$ steps. Each step is a sector with a radius of $42\text{ inches}$ and a central angle of $\frac{\pi }{8}.$

Formula Given:

The area covered by a circular sector is:

  $\text{Area}=\frac{1}{2}{r}^2\theta$

where,

  $\begin{array}{l}r=\text{radius}\\ \theta =\text{central angle }\end{array}$

Calculation:

Now, there are a total of $15$ sectors, so the total area will be:

  $\begin{array}{l}\text{Area =15}\times \frac{1}{2}{r}^2\theta \\ \text{Area =}15\times \frac{{\left(42\right)}^2}{2}\times \left(\frac{\pi }{8}\right)\\ \text{Area =}\frac{83084.4}{16}\\ \text{Area =}5192.775\end{array}$

Hence, the $5192.775{\text{ in}}^2$ carpeting would be required to cover the $15$ steps.