Question

3 select all of the true statements.
a. π is the area of a circle of radius 1
b. π is the area of a circle of diameter 1,
c. π is the circumference of a circle of radius 1.
d. π is the circumference of a circle of diameter 1.
e. π is the constant of proportionality relating the diameter of a circle to its circumference.
f. π is the constant of proportionality relating the radius of a circle to its area.
4 a class measured the radius and circumference of various circular objects. the results are
plotted on the graph.
a. does there appear to be a proportional
relationship between the radius and
circumference of a circle? explain or
show your reasoning.
b. why might the measured radii and
circumferences not be exactly
proportional?

Explanation:

Problem 3

Step1: Recall the formula for the area of a circle

The area of a circle is \(A = \pi r^{2}\). For \(r = 1\), \(A=\pi(1)^{2}=\pi\). For a circle with diameter \(d = 1\), radius \(r=\frac{d}{2}=\frac{1}{2}\), and \(A=\pi(\frac{1}{2})^{2}=\frac{\pi}{4}\).

Step2: Recall the formula for the circumference of a circle

The circumference of a circle is \(C = 2\pi r=\pi d\). For \(r = 1\), \(C = 2\pi(1)=2\pi\). For \(d = 1\), \(C=\pi(1)=\pi\).

Step3: Recall the proportionality relationships

The formula \(C=\pi d\) shows that \(\pi=\frac{C}{d}\) (constant of proportionality between circumference \(C\) and diameter \(d\)). The formula \(A=\pi r^{2}\) is not a proportional relationship (\(y = kx\) form, here \(A=\pi r^{2}\) is a quadratic relationship)

Step1: Recall the proportional relationship formula

A proportional relationship is of the form \(y=kx\). For a circle, \(C = 2\pi r\). If we check the ratios \(\frac{C}{r}\):
For the point \((3,18)\), \(\frac{18}{3}=6\). For the point \((4,25)\), \(\frac{25}{4}=6.25\). For the point \((6,38)\), \(\frac{38}{6}\approx6.33\). For the point \((7,44)\), \(\frac{44}{7}\approx6.29\).
The theoretical value of \(2\pi\approx6.28\). The points are close to the line \(y = 2\pi x\) (since the calculated ratios are close to \(2\pi\))

Measurement errors can occur. When measuring the radius of a circular object (e.g., using a ruler), there can be human - error in aligning the ruler with the center of the circle. Also, the circular objects may not be perfect circles (slight irregularities in shape), and the measuring tools (e.g., tape measures for circumference) may have limited precision.

Answer:

A. \(π\) is the area of a circle of radius \(1\) (True, since \(A=\pi r^{2}\) and \(r = 1\) gives \(A=\pi\)).
C. \(π\) is the circumference of a circle of diameter \(1\) (True, since \(C=\pi d\) and \(d = 1\) gives \(C=\pi\)).

Problem 4a