A company logo has 4 concentric circles suppose you graph the points (diameter,circumference ) and connect them with a line. In terms of (pie), what is the slope of the resulting line.
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To find the slope of the line connecting the points (diameter, circumference) on the graph, let's consider the equation of a circle in terms of its diameter.
The diameter (d) of a circle is equal to twice its radius (r). Therefore, the equation of a circle in terms of its diameter is:
d = 2r
Additionally, we know that the circumference (c) of a circle is given by:
c = πd
Substituting the value of d from the first equation into the second equation, we can express the circumference in terms of the diameter:
c = π(2r)
c = 2Ï€r
Now, let's manipulate this equation a bit further to find the slope of the line connecting the points (diameter, circumference).
We can rewrite the equation as:
r = (1/2Ï€)c
This equation represents the relationship between the radius and the circumference. We can treat r as the independent variable and c as the dependent variable.
Since we are interested in finding the slope of the line connecting the points (diameter, circumference), we can consider the slope as the rate of change of the dependent variable (circumference) with respect to the independent variable (diameter).
The slope can be calculated by taking the derivative of the equation with respect to the independent variable (diameter).
df(c)/df(d) = d((1/2Ï€)c) / d(d)
Simplifying the derivative:
df(c)/df(d) = (1/2Ï€)d(c) / d(d)
Applying the chain rule:
df(c)/df(d) = (1/2Ï€)(dc/dd)
Since the derivative of d with respect to d is 1, the equation further simplifies to:
df(c)/df(d) = (1/2Ï€)(dc/dd) = (1/2Ï€)(1)
Hence, the slope of the line connecting the points (diameter, circumference) is 1/2π, which can be written as π/2π.
Finally, canceling out π in the numerator and denominator, the slope simplifies to 1/2.
Therefore, the slope of the resulting line in terms of π is 1/2.
they will form a straight line, passing through the origin.
let the radius of the first one be r
then the first point is (2r, 2Ï€r)
slope of line passing through (0,0) and (2r, 2Ï€r)
= (2Ï€r)/(2r)
= π