One cubic meter (1.00 m3) of aluminum has a mass of 2.70E+3kg, and 1.00m3 of silver has a mass of 1.05E+4kg. Find the radius of an aluminum sphere whose mass is the same as that of an silver sphere of radius 1.07cm.
You can figure out the mass of silver, then do all the conversions of mass to volume and back, or you can note that
density(Al)/density(Ag) = 2.57
So, volume(Al)/volume(Ag) = 2.57
But, the volume of a sphere scales as the cube of the radius, so the radius scales as the cube root of the volume.
radius(Al)/radius(Ag) = 1/∛2.57 = 1.07/1.37 = 0.78 cm
Well, let's start by finding the volume of the silver sphere. The formula for the volume of a sphere is (4/3) * π * r^3. So, plugging in the given radius of 1.07cm, we have:
Silver sphere volume = (4/3) * π * (1.07cm)^3
Now let's find the mass of the silver sphere. We know that 1.00m^3 of silver has a mass of 1.05E+4kg, so we can set up the equation:
Silver mass = (1.05E+4kg / 1.00m^3) * Silver sphere volume
Now, let's find the mass of the aluminum sphere. We can set up a similar equation, using the given mass of 1.00m^3 of aluminum:
Aluminum mass = (2.70E+3kg / 1.00m^3) * Aluminum sphere volume
Since we want the mass of the aluminum sphere to be the same as the silver sphere, we can set the two mass equations equal to each other:
(1.05E+4kg / 1.00m^3) * Silver sphere volume = (2.70E+3kg / 1.00m^3) * Aluminum sphere volume
Now, we can solve for the volume of the aluminum sphere:
Silver sphere volume * (1.05E+4kg / 1.00m^3) = Aluminum sphere volume * (2.70E+3kg / 1.00m^3)
We can cancel out the units and solve for the volume:
Silver sphere volume * 1.05E+4 = Aluminum sphere volume * 2.70E+3
Now, we can plug in the value of the silver sphere volume we calculated earlier to find the volume of the aluminum sphere:
Silver sphere volume * 1.05E+4 = Aluminum sphere volume * 2.70E+3
(4/3) * π * (1.07cm)^3 * 1.05E+4 = Aluminum sphere volume * 2.70E+3
Now we just need to solve for the radius of the aluminum sphere. This can be done by rearranging the equation for the volume of a sphere:
Aluminum sphere volume = (4/3) * π * r^3
We know the volume of the aluminum sphere, so we can plug that in and solve for r:
(4/3) * π * r^3 = Silver sphere volume * 1.05E+4 / 2.70E+3
Now you can solve for r and find the radius of the aluminum sphere. Good luck, Einstein!
To find the radius of an aluminum sphere whose mass is the same as that of a silver sphere of radius 1.07 cm, we can use the known densities of aluminum and silver and the formula for the volume of a sphere.
Step 1: Convert the radius of the silver sphere to meters.
1.07 cm = 0.0107 m
Step 2: Calculate the volume of the silver sphere.
The formula for the volume of a sphere is V = (4/3)πr^3.
V_silver = (4/3)π(0.0107 m)^3
Step 3: Calculate the mass of the silver sphere using its volume and density.
Density_silver = 1.05E+4 kg / 1.00 m^3
Mass_silver = Density_silver * V_silver
Step 4: Calculate the radius of the aluminum sphere with the same mass.
Use the equation from step 2 to solve for the radius (r) in terms of mass (m):
V_aluminum = (4/3)πr^3
Density_aluminum = Mass_aluminum / 1.00 m^3
V_aluminum = Mass_aluminum / Density_aluminum
Set the volumes of the aluminum and silver spheres equal to each other:
Mass_aluminum / Density_aluminum = Mass_silver / Density_silver
Solve for Mass_aluminum:
Mass_aluminum = (Density_aluminum / Density_silver) * Mass_silver
Substitute Mass_aluminum in the equation for the volume of the aluminum sphere:
(Density_aluminum / Density_silver) * Mass_silver = (4/3)πr^3
Solve for r:
r = ((3 / 4π) * (Density_aluminum / Density_silver) * Mass_silver)^(1/3)
Step 5: Calculate the radius of the aluminum sphere.
Substitute the known values and calculate:
r = ((3 / (4 * 3.1416)) * (2.70E+3 kg / 1.05E+4 kg) * (1.05E+4 kg))^(1/3)
Calculating the expression gives:
r ≈ 0.0799 m
Therefore, the radius of the aluminum sphere whose mass is the same as that of the silver sphere with a radius of 1.07 cm is approximately 0.0799 meters.
To find the radius of the aluminum sphere, we can use the concept of density. Density is defined as the mass per unit volume. Therefore, we can set up the following equation:
Density of Aluminum = Mass of Aluminum / Volume of Aluminum
We know the mass of the aluminum (2.70E+3 kg) and the volume of the sphere can be calculated using the formula for the volume of a sphere (V = 4/3 * π * r^3, where r is the radius). Rearranging the equation, we get:
Volume of Aluminum = (Mass of Aluminum / Density of Aluminum)
Once we have the volume of the aluminum sphere, we can set it equal to the volume of the silver sphere:
Volume of Aluminum = Volume of Silver
Using the volume of the silver sphere (which is calculated using its radius), we can solve for the radius of the aluminum sphere. Let's break down the steps:
Step 1: Calculate the volume of the silver sphere.
The formula for the volume of a sphere is V = 4/3 * π * r^3. Given the radius of the silver sphere (1.07 cm), we can convert it to meters (1 m = 100 cm) and then substitute it into the formula to find the volume of the silver sphere.
Step 2: Use the known mass and density of aluminum to find the volume of the aluminum sphere.
Given the mass of the aluminum (2.70E+3 kg) and its density (which you can look up), use the equation Density = Mass / Volume to calculate the volume of the aluminum sphere.
Step 3: Set the volumes of the aluminum and silver spheres equal to each other and solve for the radius of the aluminum sphere.
Set the volume of the aluminum sphere (calculated in step 2) equal to the volume of the silver sphere (calculated in step 1) and solve for the radius of the aluminum sphere using the equation V = 4/3 * π * r^3.