Question

problem #1 - circles (mild)
complete the table. explain or show your reasoning.

problem #2 - percentages (medium)

Explanation:

The formula for the circumference \( C \) of a circle is \( C = 2\pi r \) or \( C=\pi d \) (where \( d = 2r \) is the diameter). We can use \( C = 2\pi r \) to find the circumference when the radius is known, and \( r=\frac{C}{2\pi} \) to find the radius when the circumference is known. We'll use \( \pi\approx3.14 \) for calculations.

Step 1: Ceiling Fan (Find Circumference)

Given radius \( r = 2.8 \) ft.
Using \( C = 2\pi r \), substitute \( r = 2.8 \) and \( \pi\approx3.14 \):
\( C = 2\times3.14\times2.8 \)
First, calculate \( 2\times3.14 = 6.28 \)
Then, \( 6.28\times2.8 = 17.584 \) ft.

Step 2: Water Bottle Cap (Find Circumference)

Given radius \( r = 13 \) mm.
Using \( C = 2\pi r \), substitute \( r = 13 \) and \( \pi\approx3.14 \):
\( C = 2\times3.14\times13 \)
First, \( 2\times3.14 = 6.28 \)
Then, \( 6.28\times13 = 81.64 \) mm.

Step 3: Bowl (Find Radius)

Given circumference \( C = 56.5 \) cm.
Using \( r=\frac{C}{2\pi} \), substitute \( C = 56.5 \) and \( \pi\approx3.14 \):
\( r=\frac{56.5}{2\times3.14} \)
First, calculate \( 2\times3.14 = 6.28 \)
Then, \( r=\frac{56.5}{6.28}\approx9.0 \) cm (rounded to one decimal place, or more precisely \( \frac{56.5}{6.28}\approx9.0 \) as \( 6.28\times9 = 56.52 \), very close to 56.5).

Step 4: Drum (Find Radius)

Given circumference \( C = 75.4 \) in.
Using \( r=\frac{C}{2\pi} \), substitute \( C = 75.4 \) and \( \pi\approx3.14 \):
\( r=\frac{75.4}{2\times3.14} \)
First, \( 2\times3.14 = 6.28 \)
Then, \( r=\frac{75.4}{6.28}\approx12.0 \) in (since \( 6.28\times12 = 75.36 \), very close to 75.4).

Now, we can fill the table:

object	radius	circumference
water bottle cap	13 mm	81.64 mm
bowl	~9.0 cm	56.5 cm
drum	~12.0 in	75.4 in

Answer:

The completed table (with calculated values) is as shown above. The circumferences and radii are:

• Ceiling fan circumference: \( \boldsymbol{17.584} \) ft
• Water bottle cap circumference: \( \boldsymbol{81.64} \) mm
• Bowl radius: \( \boldsymbol{\approx9.0} \) cm
• Drum radius: \( \boldsymbol{\approx12.0} \) in