Suppose that the spool is 1m long and has an inner diameter of 20 cm and an outer diameter of 60 cm. Approximately, how long a piece of 5 cm thick rope can be wound onto this spool? ( Assume that the rope is wound neatly, in layers. Each layer will consist of a row of "winds" and each "wind" will be approximately a circle.) Explain.
To determine the length of the rope that can be wound onto the spool, we need to calculate the length of each layer and then sum them up.
First, let's calculate the length of each layer. Each layer will consist of a row of "winds" that resemble approximate circles. To find the length of each wind, we need to calculate the circumference of the circle formed by the average of the inner and outer diameters.
The average diameter of the spool can be calculated as (inner diameter + outer diameter) / 2.
Therefore, the average diameter = (20 cm + 60 cm) / 2 = 40 cm.
The circumference of each wind can be calculated using the formula C = π * d, where C is the circumference and d is the diameter.
Therefore, the circumference of each wind = π * 40 cm.
To find the length of each layer, we need to multiply the circumference of each wind by the number of winds in a layer. Since the layer consists of a row of winds, the length of each layer is equal to the circumference of each wind.
Now, to find the number of winds in a layer, we need to divide the total length of the spool (1 m) by the thickness of each wind (5 cm).
Number of winds in a layer = Total spool length / Thickness of each wind
= 1 m / 0.05 m
= 20 winds.
Therefore, the length of each layer = 20 * circumference of each wind.
Finally, to find the total length of rope that can be wound onto the spool, we need to sum up the lengths of all the layers. Since the rope is wound neatly in layers, each subsequent layer will be on top of the previous one.
To summarize the steps:
1. Calculate the average diameter: (inner diameter + outer diameter) / 2.
2. Calculate the circumference of each wind: π * average diameter.
3. Calculate the number of winds in a layer: Total spool length / Thickness of each wind.
4. Calculate the length of each layer: Number of winds in a layer * circumference of each wind.
5. Sum up the length of all layers to find the total length of the rope.
By following these steps, you can accurately determine the length of the rope that can be wound onto the given spool.