Question

1. the table shows the diameters of 4 different coins.

a. complete the table.
b. to determine how much metal is on one face of a coin,
it is more useful to use the area rather than the
circumference. explain why this is the case.

Explanation:

Step1: Recall the formula for the area of a circle

The area of a circle is given by the formula \(A = \pi r^{2}\), where \(r\) is the radius of the circle. Since \(r=\frac{d}{2}\) (where \(d\) is the diameter), we can also write \(A=\pi(\frac{d}{2})^{2}=\frac{\pi d^{2}}{4}\)

Step2: Calculate the area for the Nickel

For the Nickel with \(d = 2.1\) cm.

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Step3: Calculate the area for the Dime

For the Dime with \(d = 1.8\) cm.

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Step4: Calculate the area for the Quarter

For the Quarter with \(d = 2.4\) cm.

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Step5: Explain why area is more useful than circumference

The amount of metal on one face of a coin is related to the surface - area. The circumference is a one - dimensional measure (length around the circle), while the area is a two - dimensional measure (space covered by the circular face). Since the metal covers the two - dimensional face of the coin, the area (which measures the two - dimensional space) is more relevant than the circumference (which measures a one - dimensional length around the edge)

Answer:

a.

Coin	Diameter (cm)	Area (sq.cm)
Nickel	\(2.1\)	\(\frac{\pi\times(2.1)^{2}}{4}\approx 3.46\)
Dime	\(1.8\)	\(\frac{\pi\times(1.8)^{2}}{4}\approx 2.54\)
Quarter	\(2.4\)	\(\frac{\pi\times(2.4)^{2}}{4}\approx 4.52\)

b. The amount of metal on one face of a coin is related to the two - dimensional space it covers. The circumference is a one - dimensional measure (length), while the area is a two - dimensional measure (space). Since the metal covers the face (a two - dimensional region), the area is more useful.