Verified answer

I'm just a third year high school student but I believe I know the answers to your problems.

1.) The density of gold at 25°C is 1.93 x 10⁴ kg/m³. What is the diameter(in cm) of a sphere of gold that weighs 81 g?

First, we must convert the density which is in kg/m³ to g/cm³. Via dimensional analysis, we have:

Density = 19300 kg/m³ x 1000 g/kg x (1 m / 100 cm)³

Density = 19.3 g/cm³

Second, we must find the volume of the sphere by inspecting the units of the given values we have (this still uses dimensional analysis). Density is in g/cm³. Mass is in g. If we divide mass by density, we will arrive with the unit cm³ which is volume.

Volume = 81 g / 19.3 g/cm³

Volume = 4.1969 cm³

By the formula for the volume of the sphere,

Volume = (4/3)πr³

we can get the radius (via formula manipulation). The new formula is:

Radius = ³root(3V/4π)

Radius = ³root(1.0019)

Radius = 1.0006 cm

And since diameter is twice the radius,

Diameter = 2.0012 cm

Since the least number of significant digits here is 2 (in the case of 81 g), The diameter of a sphere of gold weighing 81 g at 25°C is 2.0 cm.

2.) A certain medicine in the hospital is available as a liquid in small bottles.Each bottle contains 350μL of the medicine.The density of this medicine is 3.0 x 10³ kg/m³.

A doctor wants a certain patient to take 14.5 g of this medicine for treatment. How many bottles of the medicine should the doctor give to the patient?

First, we must convert the density of the medicine to a value consistent to that of other measurements (via dimensional analysis).

Density = 3000 kg/m³ x 1000 g/kg x (1 m / 100 cm)³

Density = 3.0 g/cm³

Since 1 cc (cubic centimeter) is equal to 1 mL, then,

Density = 3.0 g/mL

After further conversions (via dimensional analysis) you will arrive at:

Density = 3.0 g/mL x (1 mL / 1000 μL)

Density = 0.003 g/μL

After some inspection, we then realize that we can multiply the volume of medicine per bottle with the density to arrive at the mass of medicine per bottle.

Mass per bottle = 0.003 g/μL x 350 μL

Mass per bottle = 1.05 g

Now we can divide the mass required with the mass per bottle (using dimensional analysis) to arrive at the number of bottles of needed.

Bottle count = 14 g / 1.05 g/bottle

Bottle count = 13.33

Since significant figures are rarely applied for items only given in whole numbers such as bottles, we shall use a fraction for the final answer.

Bottle count = 13 and one third bottles

Voila! A third year high school student solved your problems. Shame, shame, but good luck nevertheless.

first get the volume

81g = 0.081 kg

0.081 kg / (1.93 x 10⁴ kg/m³) = 4.197e-6 m³

switching to cm

4.197e-6 m³ x (100 cm/m)³ = 4197 cm³

sphere V = ⁴/₃πr³

V = ⁴/₃πr³ = 4197

r = 10.01 cm

d = 20.02 cm