A king's crown is said to be solid gold but may be made of lead and covered with gold. when it is weighed in air, the scale reads 0.475 kg. when it is submerged in water the scale reads 0.437 kg.is it solid gold? if not, what percent is mass is gold? (density of gold= 19.3x10^3 kg/m^3)
m=0.475 kg, m1 =0.437 kg,
density of lead = 11.4 * 10^3 kg/m^3
m1•g = m•g – F(buoyancy) =>
m1•g = m•g – ρ(w)•V•g,
V= (m - m1)/ρ(w) = (0.475-0.437)/1000 =3.8•10^-5 m^3.
m1•g = ρ(x) •V•g – ρ(w)•V•g ,
m•g = ρ(x) •V•g,
ρ(x) = m/V = 0.475/3.8•10^-5 =12.5•10^3 kg/m^3
(b)
19.3•10^3•x +11.4•10^3•(100-x) = 12.5•10^3•100,
19.3•x+11.4•(100-x) = 12.5•100,
7.9•x = 110,
x =13.92% ≈14%.
The crown is roughly 14% gold and 86% lead.
To solve this problem, we can use Archimedes' principle. Archimedes' principle states that when an object is immersed in a fluid, it experiences an upward buoyant force equal to the weight of the fluid it displaces.
Let's break down the problem step by step:
Step 1: Find the weight of the crown in air.
The scale reads 0.475 kg when the crown is weighed in air.
Step 2: Find the weight of the displaced water.
Using Archimedes' principle, the weight of the displaced water is equal to the buoyant force acting on the crown when it is submerged. Since we know the density of water is approximately 1000 kg/m^3, we can calculate the weight of the displaced water using the following formula:
Weight of displaced water = Volume of the crown × Density of water × Acceleration due to gravity
Step 3: Find the apparent weight of the crown in water.
The apparent weight of the crown in water can be calculated by subtracting the weight of the displaced water from the weight of the crown in air.
Step 4: Find the density of the crown.
The density of an object can be calculated using the formula:
Density = Mass / Volume
Step 5: Find the volume of the crown.
Using the formula from step 4, we can rearrange it to calculate the volume:
Volume = Mass / Density
Step 6: Find the mass of gold in the crown.
The mass of gold in the crown can be calculated by multiplying the volume of the crown by the density of gold.
Step 7: Calculate the percentage of gold in the crown.
The percentage of gold can be calculated by dividing the mass of the gold by the total mass of the crown and multiplying by 100.
Let's apply these steps to solve the problem:
Step 1: The weight of the crown in air is 0.475 kg.
Step 2: The weight of the displaced water can be calculated using the formula:
Weight of displaced water = Volume of the crown × Density of water × Acceleration due to gravity
For simplicity, we assume the acceleration due to gravity is 9.8 m/s^2.
Weight of displaced water = Volume of the crown × 1000 kg/m^3 × 9.8 m/s^2
Step 3: The apparent weight of the crown in water is:
Weight of the crown in air - Weight of displaced water
Step 4: The density of the crown can be calculated using the formula:
Density = Mass / Volume
Step 5: The volume of the crown can be calculated using the formula:
Volume = Mass / Density
Step 6: The mass of gold in the crown can be calculated by multiplying the volume of the crown by the density of gold.
Step 7: The percentage of gold can be calculated by dividing the mass of gold by the total mass of the crown and multiplying by 100.
By following these steps, you can determine if the crown is made of solid gold and find the percentage of gold mass in the crown.