Question

3 the diameter of a canadian penny is 19 millimeters. the diameter of a canadian nickel is 21 millimeters. how much greater is the area of a canadian nickel than the area of a canadian penny? show your work. 4 a circle has radius 12 millimeters. what is its area? use 3.14 for π. show your work. 5 what is the exact area of the circle below? show your work

Explanation:

Step1: Find the radius of the penny and nickel

The radius \(r\) of a circle is half of its diameter \(d\), so \(r=\frac{d}{2}\).
For the penny, \(d = 19\) mm, then \(r_{penny}=\frac{19}{2}=9.5\) mm.
For the nickel, \(d = 21\) mm, then \(r_{nickel}=\frac{21}{2} = 10.5\) mm.

Step2: Calculate the area of the penny and nickel

The area formula of a circle is \(A=\pi r^{2}\).
The area of the penny \(A_{penny}=\pi\times(9.5)^{2}=90.25\pi\) \(mm^{2}\).
The area of the nickel \(A_{nickel}=\pi\times(10.5)^{2}=110.25\pi\) \(mm^{2}\).

Step3: Find the difference in area

\(\Delta A=A_{nickel}-A_{penny}=(110.25\pi - 90.25\pi)\) \(mm^{2}\).
Substitute \(\pi = 3.14\), \(\Delta A=(110.25 - 90.25)\times3.14\).
\(110.25-90.25 = 20\), then \(\Delta A=20\times3.14=62.8\) \(mm^{2}\).

Step4: Calculate the area of the circle with radius \(r = 12\) mm

Using the formula \(A=\pi r^{2}\), substitute \(r = 12\) mm and \(\pi=3.14\).
\(A=3.14\times12^{2}=3.14\times144 = 452.16\) \(mm^{2}\).

Step5: Calculate the exact area of the circle with \(r = 20\) cm

Using the formula \(A=\pi r^{2}\), substitute \(r = 20\) cm.
\(A=\pi\times20^{2}=400\pi\) \(cm^{2}\).

Answer:

The area of the Canadian nickel is \(62.8\) \(mm^{2}\) greater than the area of the Canadian penny.
The area of the circle with radius \(12\) mm is \(452.16\) \(mm^{2}\).
The exact area of the circle with radius \(20\) cm is \(400\pi\) \(cm^{2}\).