Question

1. chris combines 2 cups of soup that is 50° f with 1 cup of water that is 150° f. the end temperature of the mixture would be .

Explanation:

Step1: Recall the heat mixing formula

When mixing substances, the heat gained by the cooler substance equals the heat lost by the warmer substance (assuming no heat loss to the environment). The formula for the final temperature \( T \) when mixing two substances is based on the principle of conservation of energy: \( m_1c_1(T - T_1)=m_2c_2(T_2 - T) \). Since the soup and water are both liquids (we can assume the specific heat \( c \) is the same for both, so \( c_1 = c_2 \) and they cancel out. Also, the mass is proportional to the volume here (since density is similar for soup and water), so we can use volumes. Let \( V_1 = 2 \) cups, \( T_1 = 50^\circ F \), \( V_2 = 1 \) cup, \( T_2 = 150^\circ F \). The formula becomes \( V_1(T - T_1)=V_2(T_2 - T) \).

Step2: Substitute the values

Substitute \( V_1 = 2 \), \( T_1 = 50 \), \( V_2 = 1 \), \( T_2 = 150 \) into the equation:
\( 2(T - 50)=1(150 - T) \)

Step3: Expand and solve for T

Expand the left side: \( 2T - 100 = 150 - T \)
Add \( T \) to both sides: \( 2T+T - 100=150 - T+T \)
\( 3T - 100 = 150 \)
Add 100 to both sides: \( 3T - 100 + 100=150 + 100 \)
\( 3T = 250 \)? Wait, no, wait: Wait, 150 + 100 is 250? Wait, no, 150 + 100 is 250? Wait, no, 150 + 100 is 250? Wait, no, 150 + 100 is 250? Wait, no, let's recalculate. Wait, 2(T - 50)=150 - T. So 2T - 100 = 150 - T. Then 2T + T = 150 + 100. 3T = 250? Wait, that can't be. Wait, maybe I made a mistake. Wait, the formula is actually the weighted average. Since the specific heat is the same, the final temperature is the weighted average of the two temperatures, weighted by their volumes (or masses). So the formula is \( T=\frac{m_1T_1 + m_2T_2}{m_1 + m_2} \). Since \( m \propto V \), we can use volumes. So \( T=\frac{V_1T_1 + V_2T_2}{V_1 + V_2} \).

Step4: Calculate using the weighted average formula

Substitute \( V_1 = 2 \), \( T_1 = 50 \), \( V_2 = 1 \), \( T_2 = 150 \):
\( T=\frac{2\times50 + 1\times150}{2 + 1}=\frac{100 + 150}{3}=\frac{250}{3}\approx83.33^\circ F \)

Wait, let's check the first equation again. \( 2(T - 50)=150 - T \). So 2T - 100 = 150 - T. Then 2T + T = 150 + 100. 3T = 250? No, 150 + 100 is 250? Wait, 150 + 100 is 250? Yes. Then T = 250/3 ≈ 83.33. That's correct.

Answer:

\(\boxed{\frac{250}{3}^\circ F}\) or approximately \(\boxed{83.33^\circ F}\) (depending on the required format, but since the problem doesn't specify, the exact fraction is \(\frac{250}{3}\) or the decimal approximation).