Question:

rewrite same as given below and strcture the answer ### **Part 1: Coffee Cooling Problem (Newton’s Law of Cooling)** Newton’s Law of Cooling is: $$ T(t) = T_{\text{room}} + (T_0 - T_{\text{room}}) e^{-kt} $$ --- **Where:** * $T(t)$: temperature at time $t$ * $T_0$: initial temperature ($95^\circ C$) * $T_{\text{room}}$: room temperature (30°C for A, 20°C for B) * $k$: cooling constant (depends on environment and properties of the liquid) --- **Given:** * At $t = 0$: $$ T_0 = 95^\circ C $$ * After 10 minutes: * $T_A = 89^\circ C$, $T_{\text{room, A}} = 30^\circ C$ * $T_B = 93^\circ C$, $T_{\text{room, B}} = 20^\circ C$ --- --- ### **a) Temperature at any time $t$** #### **For coffee A:** $$ 89 = 30 + (95 - 30) e^{-10k_A} \Rightarrow 59 = 65e^{-10k_A} \Rightarrow e^{-10k_A} = \frac{59}{65} $$ $$ k_A = -\frac{1}{10} \ln\left(\frac{59}{65}\right) \Rightarrow \boxed{k_A \approx 0.0098} $$ Thus, $$ T_A(t) = 30 + 65e^{-0.0098t} $$ --- #### **For coffee B:** $$ 93 = 20 + (95 - 20) e^{-10k_B} \Rightarrow 73 = 75e^{-10k_B} \Rightarrow e^{-10k_B} = \frac{73}{75} $$ $$ k_B = -\frac{1}{10} \ln\left(\frac{73}{75}\right) \Rightarrow \boxed{k_B \approx 0.0027} $$ Thus, $$ T_B(t) = 20 + 75e^{-0.0027t} $$ --- ### **b) After 5 hours (i.e., $t = 300$), find temperature:** #### **For A:** $$ T_A(300) = 30 + 65e^{-0.0098 \times 300} = 30 + 65e^{-2.94} \approx 30 + 65 \cdot 0.0527 \approx \boxed{33.43^\circ C} $$ --- #### **For B:** $$ T_B(300) = 20 + 75e^{-0.0027 \times 300} = 20 + 75e^{-0.81} \approx 20 + 75 \cdot 0.445 \approx \boxed{53.38^\circ C} $$ --- --- ### **c) Which coffee is cooling faster?** Coffee A is cooling faster because it has a **higher $k$ value** (0.0098 vs 0.0027), which means it loses heat more quickly. Additionally, since it is placed in a warmer environment (30°C), the temperature difference between the coffee and the surroundings is smaller. Yet it still cools more than Coffee B, confirming that Coffee A has a **higher cooling constant**. --- ### **Superposition Theorem: Overview** You are asked to **verify that $V_z = 45 \text{ V}$** in the circuit using **superposition**. This involves: * Considering **one source at a time**, turning the others off: * Replace **voltage sources** with **short circuits** * Replace **current sources** with **open circuits** --- Here's the clean, organized rewritten version of your content: --- ## ✅ **Coffee Cooling Problem** ### **a) Temperature Functions:** $$ T_A(t) = 30 + 65e^{-0.0098t} $$ $$ T_B(t) = 20 + 75e^{-0.0027t} $$ --- ### **b) Temperature After 5 Hours:** $$ T_A(300) \approx 33.43^\circ C $$ $$ T_B(300) \approx 53.38^\circ C $$ --- ### **c) Which Coffee Cools Faster?** Coffee A is cooling faster because it has a **higher cooling constant $k$**. Despite being in a warmer room (30°C), it loses heat more rapidly than Coffee B (room temperature 20°C), indicating a stronger cooling rate. --- ## ✅ **Circuit Problem – Using Superposition Theorem** **Given:** * Resistors: 10 Ω, 30 Ω, 25 Ω, 100 Ω, 50 Ω * Sources: 36 V (left), 27 V (right), 3.6 A (bottom left current source) * Objective: Find voltage $V_x$ across the 50 Ω resistor --- ### **Step 1: Only 36 V Active (Others OFF)** * Short 27 V source * Open 3.6 A source **Circuit Insight:** * 36 V drops across 10 Ω and 30 Ω in series * 25 Ω and 100 Ω are disconnected * 50 Ω is between the junction of 10Ω–30Ω and ground **Voltage Divider:** $$ V_{\text{node}} = 36 \cdot \frac{30}{10 + 30} = 36 \cdot \frac{30}{40} = 27\text{ V} $$ ✅ Contribution to $V_x = \boxed{27\text{ V}}$ --- ### **Step 2: Only 27 V Active (Others OFF)** * Short 36 V source * Open 3.6 A source **Circuit Insight:** * 27 V across 30 Ω and 50 Ω loop **Voltage Divider:** $$ V_{\text{node}} = 27 \cdot \frac{50}{30 + 50} = 27 \cdot \frac{50}{80} = 16.875\text{ V} $$ ✅ Contribution to $V_x = \boxed{16.875\text{ V}}$ --- ### **Step 3: Only 3.6 A Current Source Active (Others OFF)** * Short both voltage sources **Circuit Insight:** * Current flows through 25 Ω and 100 Ω (in series → 125 Ω) * 50 Ω connected in parallel to part of the network **After full mesh or nodal analysis (omitted here):** ✅ Contribution to $V_x = \boxed{1.125\text{ V}}$ --- ### **Final Answer – Total Voltage:** $$ V_x = 27 + 16.875 + 1.125 = \boxed{45\text{ V}} $$ ✅ Verified using **Superposition Theorem** --- annd also write final answer