Gold, which has a density of 19.32 g/cm^3, is the most ductile metal and can be pressed into a thin leaf or drawn out into a long fiber. (a) If a sample of gold with a mass of 6.710 g, is pressed into a leaf of 1.661 μm thickness, what is the area (in m^2) of the leaf? (b) If, instead, the gold is drawn out into a cylindrical fiber of radius 2.800 μm, what is the length (in m) of the fiber?

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Answer is 1

To answer these questions, we need to use the given information and the formula for calculating the area and length.

(a) To find the area of the leaf, we can use the formula:

Area = mass / (density * thickness)

Given:
- Gold density: 19.32 g/cm^3
- Mass: 6.710 g
- Thickness: 1.661 μm

We need to convert the thickness from micrometers (μm) to centimeters (cm) because the density is given in grams per cubic centimeter (g/cm^3):

1 μm = 1 × 10^-4 cm

So the thickness in cm = 1.661 μm * 1 × 10^-4 cm/μm = 1.661 × 10^-4 cm

Plugging in the values into the formula, we find:

Area = 6.710 g / (19.32 g/cm^3 * 1.661 × 10^-4 cm)
= (6.710 g * cm) / (19.32 g * 1.661 × 10^-4)

Canceling out the units, we have:

Area = 6.710 / (19.32 * 1.661 × 10^-4) cm^2

Finally, we need to convert the area from cm^2 to m^2 by dividing by 10,000 (since there are 10,000 cm^2 in 1 m^2):

Area = (6.710 / (19.32 * 1.661 × 10^-4)) / 10000 m^2

(b) To find the length of the fiber, we can use the formula:

Length = mass / (density * π * radius^2)

Given:
- Gold density: 19.32 g/cm^3
- Mass: 6.710 g
- Radius: 2.800 μm

First, we need to convert the radius from micrometers (μm) to centimeters (cm):

2.800 μm = 2.800 × 10^-4 cm

Plugging in the values into the formula, we find:

Length = 6.710 g / (19.32 g/cm^3 * π * (2.800 × 10^-4 cm)^2)

Simplifying the equation, we have:

Length = (6.710 / (19.32 * π * (2.800 × 10^-4)^2)) cm

Finally, to convert the length from cm to m, we divide by 100:

Length = ((6.710 / (19.32 * π * (2.800 × 10^-4)^2))) / 100 m

Therefore, the area of the leaf is given by the formula (6.710 / (19.32 * 1.661 × 10^-4)) / 10000 m^2, and the length of the fiber is given by the formula ((6.710 / (19.32 * π * (2.800 × 10^-4)^2))) / 100 m.