To solve the problem, we need to use the fact that \( p \) varies directly as \( T \). This means that the ratio \( \frac{p}{T} \) remains constant. Given the initial conditions, we can set up the following relationship:
[tex]\[ \frac{p_1}{T_1} = \frac{p_2}{T_2} \][/tex]
Here are the known values:
- \( p_1 = 10^3 \)
- \( T_1 = 400 \)
- \( T_2 = 500 \)
We need to find \( p_2 \), the pressure when the temperature is 500. Using the relationship above, we set up the equation:
[tex]\[ \frac{10^3}{400} = \frac{p_2}{500} \][/tex]
To isolate \( p_2 \), we multiply both sides by 500:
[tex]\[ p_2 = 10^3 \times \frac{500}{400} \][/tex]
Simplifying this:
[tex]\[ p_2 = 10^3 \times 1.25 \][/tex]
[tex]\[ p_2 = 1250 \][/tex]
Therefore, the pressure \( p \) when the temperature \( T \) is 500 is \( 1250 \).
Given the options:
(a) \( 1.5 \times 10^3 \)
(b) \( 1.5 \times 10^4 \)
The value \( 1250 \) corresponds to \( 1.25 \times 10^3 \), which is not one of the provided choices. However, the closest to our calculated value of \( 1250 \) is not clearly given in the options provided.
So, the correct answer \( p_2 = 1250 \) does not match any of the given choices exactly.
However, [tex]\( 1.25 \times 10^3 \)[/tex] would be the scientific notation for [tex]\( 1250 \)[/tex], which might be the intended answer if we consider a discrepancy in provided choices.
