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To determine the percentage error in the density [tex]\( \rho \)[/tex] of a piece of metal, given that [tex]\( m^4 = 375.32 \pm 0.01 \text{ g}^4 \)[/tex] and [tex]\( V = 136.41 \pm 0.01 \text{ cm}^3 \)[/tex], we can follow these steps:
### 1. Calculate the Mass [tex]\( m \)[/tex]
First, find the mass [tex]\( m \)[/tex] from [tex]\( m^4 \)[/tex]:
[tex]\[ m^4 = 375.32 \implies m = (375.32)^{1/4} \][/tex]
The numerical result is:
[tex]\[ m = 4.4014971695575245 \text{ g} \][/tex]
### 2. Determine the Error in [tex]\( m \)[/tex]
Using propagation of uncertainty, the error in [tex]\( m \)[/tex] ([tex]\(\Delta m\)[/tex]) can be found from the given error in [tex]\( m^4 \)[/tex]:
[tex]\[ \Delta m = \left| \frac{\partial m}{\partial m^4} \right| \Delta m^4 \][/tex]
Where:
[tex]\[ \left| \frac{\partial m}{\partial m^4} \right| = \left| \frac{1}{4} (m^4)^{-3/4} \right| = \frac{1}{4} (375.32)^{-3/4} \][/tex]
Plugging in the numbers, we get:
[tex]\[ \Delta m = 0.25 \times (375.32)^{-0.75} \times 0.01 = 2.9318296184306223\times 10^{-5} \text{ g} \][/tex]
### 3. Calculate the Density [tex]\( \rho \)[/tex]
The density [tex]\( \rho \)[/tex] is calculated using:
[tex]\[ \rho = \frac{m}{V} = \frac{4.4014971695575245 \text{ g}}{136.41 \text{ cm}^3} = 0.03226667524050674 \text{ g/cm}^3 \][/tex]
### 4. Determine the Error in [tex]\( \rho \)[/tex]
To find the error in [tex]\( \rho \)[/tex] ([tex]\(\Delta \rho\)[/tex]), we use the propagation of uncertainty formula for a quotient:
[tex]\[ \Delta \rho = \rho \sqrt{\left( \frac{\Delta m}{m} \right)^2 + \left( \frac{\Delta V}{V} \right)^2} \][/tex]
Where [tex]\(\Delta V = 0.01 \text{ cm}^3\)[/tex]. Plugging in the values:
[tex]\[ \Delta \rho = 0.03226667524050674 \sqrt{\left( \frac{2.9318296184306223\times 10^{-5}}{4.4014971695575245} \right)^2 + \left( \frac{0.01}{136.41} \right)^2} = 2.3751629707308275\times 10^{-6} \text{ g/cm}^3 \][/tex]
### 5. Calculate the Percentage Error in [tex]\( \rho \)[/tex]
Finally, the percentage error in [tex]\( \rho \)[/tex] is given by:
[tex]\[ \%\text{ error in } \rho = \left( \frac{\Delta \rho}{\rho} \right) \times 100 \][/tex]
Substituting the values we found:
[tex]\[ \%\text{ error in } \rho = \left( \frac{2.3751629707308275\times 10^{-6}}{0.03226667524050674} \right) \times 100 = 0.007361040308699392 \% \][/tex]
Thus, the percentage error in [tex]\( \rho \)[/tex] is approximately [tex]\( 0.00736\% \)[/tex].
### 1. Calculate the Mass [tex]\( m \)[/tex]
First, find the mass [tex]\( m \)[/tex] from [tex]\( m^4 \)[/tex]:
[tex]\[ m^4 = 375.32 \implies m = (375.32)^{1/4} \][/tex]
The numerical result is:
[tex]\[ m = 4.4014971695575245 \text{ g} \][/tex]
### 2. Determine the Error in [tex]\( m \)[/tex]
Using propagation of uncertainty, the error in [tex]\( m \)[/tex] ([tex]\(\Delta m\)[/tex]) can be found from the given error in [tex]\( m^4 \)[/tex]:
[tex]\[ \Delta m = \left| \frac{\partial m}{\partial m^4} \right| \Delta m^4 \][/tex]
Where:
[tex]\[ \left| \frac{\partial m}{\partial m^4} \right| = \left| \frac{1}{4} (m^4)^{-3/4} \right| = \frac{1}{4} (375.32)^{-3/4} \][/tex]
Plugging in the numbers, we get:
[tex]\[ \Delta m = 0.25 \times (375.32)^{-0.75} \times 0.01 = 2.9318296184306223\times 10^{-5} \text{ g} \][/tex]
### 3. Calculate the Density [tex]\( \rho \)[/tex]
The density [tex]\( \rho \)[/tex] is calculated using:
[tex]\[ \rho = \frac{m}{V} = \frac{4.4014971695575245 \text{ g}}{136.41 \text{ cm}^3} = 0.03226667524050674 \text{ g/cm}^3 \][/tex]
### 4. Determine the Error in [tex]\( \rho \)[/tex]
To find the error in [tex]\( \rho \)[/tex] ([tex]\(\Delta \rho\)[/tex]), we use the propagation of uncertainty formula for a quotient:
[tex]\[ \Delta \rho = \rho \sqrt{\left( \frac{\Delta m}{m} \right)^2 + \left( \frac{\Delta V}{V} \right)^2} \][/tex]
Where [tex]\(\Delta V = 0.01 \text{ cm}^3\)[/tex]. Plugging in the values:
[tex]\[ \Delta \rho = 0.03226667524050674 \sqrt{\left( \frac{2.9318296184306223\times 10^{-5}}{4.4014971695575245} \right)^2 + \left( \frac{0.01}{136.41} \right)^2} = 2.3751629707308275\times 10^{-6} \text{ g/cm}^3 \][/tex]
### 5. Calculate the Percentage Error in [tex]\( \rho \)[/tex]
Finally, the percentage error in [tex]\( \rho \)[/tex] is given by:
[tex]\[ \%\text{ error in } \rho = \left( \frac{\Delta \rho}{\rho} \right) \times 100 \][/tex]
Substituting the values we found:
[tex]\[ \%\text{ error in } \rho = \left( \frac{2.3751629707308275\times 10^{-6}}{0.03226667524050674} \right) \times 100 = 0.007361040308699392 \% \][/tex]
Thus, the percentage error in [tex]\( \rho \)[/tex] is approximately [tex]\( 0.00736\% \)[/tex].
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