Considering the area of the hole, it is found that there is a 4.9% probability of hitting a ball into the circular hole.
<h3>What is the area of a circle?</h3>The area of a circle of radius r is given by:
In this problem, the hole has a diameter of 1 ft, hence it is radius is of r = 1 ft/2 = 0.5 ft, and it's area is given by:
Hence the probability is given by:
p = 0.7854/16 = 0.049 = 4.9%
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In the graph, the constant of proportionality is 40
<h3>Calculating the constant of proportionality</h3>From the question, we are to determine the constant of proportionality
The constant of proportionality =
From the graph, a point on the line is (1, 40)
That is,
When x = 1 and y = 40
Thus,
The constant of proportionality =
Constant of proportionality = 40
Hence, the constant of proportionality is 40
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Let
where we assume |r| < 1. Multiplying on both sides by r gives
and subtracting this from gives
As n → ∞, the exponential term will converge to 0, and the partial sums will converge to
Now, we're given
We must have |r| < 1 since both sums converge, so
Solving for r by substitution, we have
Recalling the difference of squares identity, we have
We've already confirmed r ≠ 1, so we can simplify this to
It follows that
and so the sum we want is
which doesn't appear to be either of the given answer choices. Are you sure there isn't a typo somewhere?