Answer:
a) 16675.75 Kg/m³ b) 77.6%
Explanation:
the weight of the crown = 60 N, density of gold = 19300 Kg/m^3, density of lead = 11340 kg/m^3, density of water = 1000kg/m^3 and acceleration due to gravity = 9.8 m/s^2
upthrust on the crown = weight in air - weight when fully submerged in water = 60 - 56.4 = 3.6 N
mass of water displaced = 3.6 / 9.8 since weight = mass × g
mass of water displaced = 0.367 Kg
density of water = mass / volume
1000 = 0.367 / volume
cross multiply and find volume
volume of the crown = 0.367 / 1000 = 0.000367 m³ since the crown will displace water of equal volume according to Archimedes principle
Let V1 represent the volume of Gold and let V2 represent the volume of lead
Total volume of the crown = V1 + V2
also
density of gold = mass of gold / V1 and density of lead = mass of lead / V2
19300 = mass of gold in the crown / V1 and 11340 = mass of lead in the crown / V2
19300 V1 = mass of gold and 11340 V2 = mass of lead
add the two together
19300 V1 + 11340 V2 = weigth of the crown / 9.8
19300 V1 + 11340 V2 = 6.12 also
V1 + V2 = 0.000367
make V1 subject of the formula in equation 2
V1 = 0.000367 - V2
substitute for V1 in equation 1
19300 (0.000367 - V2) + 11340 V2 = 6.12
open the bracket
7.083 - 19300 V2 + 11340 V2 = 6.12
rearrange the equation
-7960 V2 = 6.12 - 7.083
-7960 V2 = -0.963
V2 = -0.963 / -7960 = 0.000121 (volume of lead in the crown)
substitute V2 into equation 2
V1 + 0.000121 = 0.000367m³
V1 = 0.000367 - 0.000121 = 0.000246m³ (volume of gold in the crown)
so mass of gold in the crown = 19300 × 0.000246 = 4.748 kg
and mass of lead = 11340 × 0.000121 = 1.372 kg
average density of the crown = (mass of gold + mass of lead) / total volume = 6.12 / 0.000367 = 16675.75 kg/ m³
b) percentage make of gold = mass of gold / total mass × 100 = 77.6 % approx
