The figure shows the beginning of a spiral created by starting with a semicircle
of radius 1 unit and endlessly attaching semicircles that are each
half the radius of the previous semicircle. What is the length of
the whole spiral? Express your answer in terms of π.
The length of each semicircle can be calculated using the formula for the circumference of a circle:
C = πd
Since the radius of the first semicircle is 1 unit, its diameter is 2 units. Therefore, the length of the first semicircle is:
C1 = π(2) = 2π
The radius of the second semicircle is half of the radius of the first semicircle, so it is 1/2 units. Therefore, the length of the second semicircle is:
C2 = π(1) = π
Similarly, the radius of the third semicircle is half of the radius of the second semicircle, so it is 1/4 units. Therefore, the length of the third semicircle is:
C3 = π·(1/2) = π/2
This pattern continues, with each semicircle having a radius that is half of the previous semicircle:
C4 = π·(1/4) = π/4
C5 = π·(1/8) = π/8
C6 = π·(1/16) = π/16
...
To find the length of the entire spiral, we need to add up all these lengths. Since the lengths form a geometric series with a common ratio of 1/2, we can use the formula for the sum of an infinite geometric series:
S = a / (1 - r)
where S is the sum of the series, a is the first term, and r is the common ratio. In this case, a = 2π and r = 1/2, so we have:
S = (2π) / (1 - 1/2)
S = (2π) / (1/2)
S = 4π
Therefore, the length of the whole spiral is 4π units.