Answer:
a) 16675.75 Kg/m³ b) 77.6%
Explanation:
The weight of the crown is 60 N, with gold's density at 19300 Kg/m³, lead's at 11340 kg/m³, water's at 1000kg/m³, and gravitational acceleration at 9.8 m/s².
The upthrust acting on the crown equals the weight in air minus its weight in water: 60 - 56.4 results in 3.6 N.
This leads to the mass of water displaced being 3.6 / 9.8, as weight equals mass times gravity.
The mass of displaced water is 0.367 Kg.
The density of water relates mass to volume as: 1000 = 0.367 / volume.
Cross-multiplying helps us determine the volume:
The crown’s volume becomes 0.367 / 1000 = 0.000367 m³ since it displaces an equal volume of water per Archimedes' principle.
Let V1 denote the gold volume and V2 the lead volume.
Total volume for the crown becomes V1 + V2.
Likewise, using the relationship of densities:
Density of gold translates to mass of gold over V1 and lead density translates to mass of lead over V2.
Thus, 19300 = mass of gold in the crown / V1 and 11340 = mass of lead in the crown / V2.
Combining them gives us: 19300 V1 = mass of gold and 11340 V2 = mass of lead.
Adding together leads to: 19300 V1 + 11340 V2 = weight of the crown / 9.8.
So, 19300 V1 + 11340 V2 = 6.12.
From V1 + V2 = 0.000367, we can express V1 in relation to V2.
V1 = 0.000367 - V2.
Substituting this into the mass equation results in:
19300 (0.000367 - V2) + 11340 V2 = 6.12.
Expanding gives:
7.083 - 19300 V2 + 11340 V2 = 6.12.
Reorganizing yields:
-7960 V2 = 6.12 - 7.083.
So, -7960 V2 = -0.963.
This leads to V2 = -0.963 / -7960 = 0.000121 (the lead volume in the crown).
Substituting V2 back into the total volume equation gives:
V1 + 0.000121 = 0.000367 m³
Thus, V1 = 0.000367 - 0.000121 = 0.000246 m³ (the gold volume in the crown).
Which leads to the mass of gold in the crown = 19300 × 0.000246 = 4.748 kg.
The mass of lead equals 11340 × 0.000121 = 1.372 kg.
The average density for the crown calculates as (mass of gold + mass of lead) / total volume = 6.12 / 0.000367 = 16675.75 kg/m³.
b) The percentage of gold by weight computes to mass of gold / total mass × 100 = approximately 77.6%.
