To determine the change in volume of the right rectangular prism due to the change in temperature, we will use the formula for volumetric expansion, which is given by:
\[ \Delta V = V_0 \cdot \beta \cdot \Delta T \]
Where:
- \( \Delta V \) is the change in volume,
- \( V_0 \) is the original volume,
- \( \beta \) is the coefficient of volumetric expansion,
- \( \Delta T \) is the change in temperature.
The coefficient of volumetric expansion (\( \beta \)) for isotropic materials can be approximated as:
\[ \beta = 3 \cdot \alpha \]
Where \( \alpha \) is the coefficient of linear expansion.
Given:
- \( \alpha = 14 \times 10^{-6} , \text{°C}^{-1} \)
- \( \Delta T = 45 , \text{°C} \)
We can calculate \( \beta \):
\[ \beta = 3 \cdot (14 \times 10^{-6} , \text{°C}^{-1}) = 42 \times 10^{-6} , \text{°C}^{-1} \]
Next, we need to calculate the original volume \( V_0 \) of the prism:
\[ V_0 = \text{length} \times \text{width} \times \text{height} = 70 , \text{mm} \times 130 , \text{mm} \times 10 , \text{mm} \]
Calculating this:
\[ V_0 = 70 \times 130 \times 10 = 91000 , \text{mm}^3 \]
Now we can use the original volume, volumetric expansion coefficient, and change in temperature to find \( \Delta V \):
\[ \Delta V = 91000 , \text{mm}^3 \cdot (42 \times 10^{-6} , \text{°C}^{-1}) \cdot 45 , \text{°C} \]
Calculating this step by step:
- Calculate the product of \( 42 \times 10^{-6} \) and \( 45 \):
\[ 42 \times 10^{-6} \cdot 45 = 1.89 \times 10^{-3} \]
- Now calculate \( \Delta V \):
\[ \Delta V = 91000 , \text{mm}^3 \cdot 1.89 \times 10^{-3} = 172.59 , \text{mm}^3 \]
Thus, the change in volume \( \Delta V \) is approximately:
\[ \Delta V \approx 172.6 , \text{mm}^3 \]
Final answer: The change in volume is approximately 172.6 mm³.