Shown in the figure are circles of diameter 42 cm. What is the total length (in cm) of the path LRMONP as indicated by the red arrows? Assume \(\pi = \frac{22}{7}\).
Show Hint
When calculating the length of a path involving circular arcs, use the formula for the arc length and add the straight segments to get the total length.
Step 1: Understanding the figure.
We are given four circles with a diameter of 42 cm each. The path LRMONP consists of straight lines and circular arcs along the circumference of the circles.
Step 2: Calculating the radius of the circles.
The radius \( r \) of each circle is half of the diameter:
\[
r = \frac{42}{2} = 21 \, \text{cm}
\]
Step 3: Length of the circular arcs.
The length of a circular arc is given by the formula:
\[
\text{Length of arc} = \theta \times r
\]
where \( \theta \) is the central angle in radians.
- From the diagram, the path \( LR \) and \( NP \) represent quarter-circle arcs, i.e., \( \theta = \frac{\pi}{2} \). Thus, the length of each quarter-circle arc is:
\[
\text{Length of arc} = \frac{\pi}{2} \times 21 = \frac{22}{7} \times \frac{1}{2} \times 21 = 33 \, \text{cm}
\]
Since there are two arcs, the total length of the arcs is:
\[
33 + 33 = 66 \, \text{cm}
\]
Step 4: Length of the straight segments.
The straight segments \( LM \) and \( ON \) are equal to the diameter of the circles, which is 42 cm.
Step 5: Total length of the path.
The total length of the path \( LRMONP \) is the sum of the lengths of the straight segments and the arcs:
\[
\text{Total length} = 42 + 42 + 66 = 150 \, \text{cm}
\]
\[
\boxed{150 \, \text{cm}}
\]
