To solve the problem of determining the length of the rectangle, we can follow these steps:
1. Calculate the circumference of the circle:
The circumference of a circle is given by the formula:
[tex]\[
\text{Circumference} = 2 \pi r
\][/tex]
where [tex]\( r \)[/tex] is the radius of the circle.
Here, the radius [tex]\( r \)[/tex] is 27 inches.
So, we have:
[tex]\[
\text{Circumference} = 2 \times \pi \times 27 \approx 169.65 \text{ inches}
\][/tex]
2. Relate the circumference to the perimeter of the rectangle:
The length of the rope (which is the circumference of the circle) is now the perimeter of the rectangle. The perimeter [tex]\( P \)[/tex] of a rectangle is given by the formula:
[tex]\[
P = 2(\text{Length} + \text{Width})
\][/tex]
Since the perimeter of the rectangle must equal the circumference of the circle, we set the perimeter equal to 169.65 inches.
3. Setup the equation and solve for the length of the rectangle:
Let's denote the length of the rectangle by [tex]\( L \)[/tex] and the width by [tex]\( W \)[/tex]. Given that [tex]\( W = 17 \)[/tex] inches, we can write:
[tex]\[
2(L + 17) = 169.65
\][/tex]
4. Isolate [tex]\( L \)[/tex] and solve:
[tex]\[
L + 17 = \frac{169.65}{2} \approx 84.825
\][/tex]
[tex]\[
L = 84.825 - 17 \approx 67.825
\][/tex]
5. Round the length to the nearest hundredth:
[tex]\[
L \approx 67.82
\][/tex]
Therefore, the length of the rectangle, rounded to the nearest hundredth of an inch, is [tex]\( \boxed{67.82} \)[/tex].
So, the correct answer is:
- E. 67.82
