Question

A plastic cover rests on the top of a pool and includes a wedge-shaped flap that forms a ${45}^{\circ }$ angle at the centre of the cover, as shown in the figure. A zipper goes along $1$ side of the wedge-shaped flap and around the arc. Which of the following is closest to the length, in feet, of the zipper if the length of arc made by the flap is 9.42 ft?(Use $\pi =3.14$)

• A. $24$
• B. $21.42$
• C. $10.71$
• D. $18.84$

Answer 1

The correct option is B $21.42$

Since $45$ degrees is $\frac{1}{8}$ the circumference. Therefore, the circumference is $2\pi r$.

Thus, $\frac{1}{8}$th would be $\frac{2\pi r}{8}$

The length of the zipper from the center of the pool will be
$L=r+\left(\frac{2\pi r}{8}\right)$
Now the length of the arc is $9.42$ feet and it is $\frac{1}{8}$th of the circumference then we can calculate the radius
$C=2\pi r$

$9.42\times 8=2\pi r$

$r=9.42\times \frac{8}{2\times 3.14}=12$

Therefore, $r=12$ to the nearest value.
The zipper length is $12+9.42=21.42$ feet.