Suppose a metal worker shapes a 3.4 mg piece of gold until it is a thin, perfectly square plate that is uniformly 8.6x10^-6 cm thick. What is the length of a side of the plate, in cm?
mass = 3.4 mg = 0.0034 grams.
volume Au = mass/density. You will need to look up the density of Au.
Then volume = length x width x thickness. You are given the thickness, you've calculated the volume, left is length and width but since that is a square, you solve for that.
That is volume = side x side x thickness. Solve for side.
To find the length of a side of the square plate, we can divide the mass of the gold by its density and then calculate its volume and subsequently its side length using the formula for the volume of a square plate.
Here's how you can calculate it step by step:
Step 1: Calculate the volume of the gold plate
Density is defined as mass per unit volume, so we can use the formula: Density = Mass / Volume.
Given that the mass of the gold plate is 3.4 mg and the density of gold is 19.3 g/cm³, we need to convert the mass to grams:
3.4 mg = 3.4 × 10⁻³ g
Now, we can rearrange the density formula to solve for volume:
Volume = Mass / Density
Volume = (3.4 × 10⁻³ g) / (19.3 g/cm³)
Step 2: Calculate the side length of the square plate
The volume of a square plate is given by the formula: Volume = side length × side length × thickness.
We know the volume from Step 1 and the thickness of the plate is 8.6 × 10⁻⁶ cm.
Volume = side length × side length × thickness
(3.4 × 10⁻³ g) / (19.3 g/cm³) = side length × side length × (8.6 × 10⁻⁶ cm)
Now, we can rearrange the equation to solve for the side length:
side length = √((3.4 × 10⁻³ g) / (19.3 g/cm³)) / (8.6 × 10⁻⁶ cm)
Calculating this equation will give us the length of a side of the plate in cm.