https://file.aiquickdraw.com/mathbot/file/1727521412561lzj5dca6.jpg

Let's solve the three problems based on the image.

### 1. Lead Heating Problem
**Given:**
- Mass of lead \( m = 500 \, \text{g} \)
- Initial temperature \( T_1 = 25^\circ \, \text{C} \)
- Final temperature \( T_2 = 75^\circ \, \text{C} \)
- Specific heat of lead \( c = 0.129 \, \text{J/g}^\circ \text{C} \)

**Formula:**
The energy required to heat the lead is given by:
\[
Q = mc\Delta T
\]
Where \( \Delta T = T_2 - T_1 \).

### 2. Specific Heat of Metal Problem
**Given:**
- Mass of the metal ball \( m = 25 \, \text{g} \)
- Temperature change \( \Delta T = 200^\circ \, \text{C} \) (assuming from 0°C to 200°C)
- Energy used \( Q = 2330 \, \text{J} \)

**Formula:**
To find the specific heat \( c \), we rearrange the formula:
\[
Q = mc\Delta T \quad \text{or} \quad c = \frac{Q}{m\Delta T}
\]

### 3. Copper Cooling Problem
**Given:**
- Mass of copper \( m = 1 \, \text{kg} = 1000 \, \text{g} \)
- Final temperature \( T_2 = 100^\circ \, \text{C} \)
- Energy released \( Q = 231 \, \text{kJ} = 231,000 \, \text{J} \)
- Specific heat of copper \( c = 0.385 \, \text{J/g}^\circ \text{C} \)

**Formula:**
We use the same formula, but rearrange to find the initial temperature \( T_1 \):
\[
Q = mc(T_1 - T_2) \quad \text{or} \quad T_1 = \frac{Q}{mc} + T_2
\]

Now let's calculate these values.Here are the results for the three problems:

1. **Lead Heating Problem:**
   The energy required to heat the lead from \( 25^\circ C \) to \( 75^\circ C \) is:
   \[
   Q = 3225 \, \text{J}
   \]

2. **Specific Heat of the Metal:**
   The specific heat of the metal ball is:
   \[
   c = 0.466 \, \text{J/g}^\circ C
   \]

3. **Copper Cooling Problem:**
   The initial temperature of the copper before it cooled to \( 100^\circ C \) was:
   \[
   T_1 = 700^\circ C
   \]

Would you like more details or have any questions?

Here are 5 related questions:
1. How does the specific heat of a material affect its heating rate?
2. Can the energy calculated in the first problem be converted into other units like calories?
3. How do we account for heat losses in real-world heating problems?
4. Why is the specific heat of metals generally lower than that of water?
5. How do you measure specific heat in a laboratory setting?

**Tip:** When heating or cooling materials, always ensure you're working in consistent units (e.g., mass in grams, energy in joules).

1. A 500-gram cube of lead is heated from 25°C to 75°C. How much energy was required to heat the lead? The specific heat of lead is 0.129 J/g°C. 2. A 25-gram metal ball is heated to 200°C with 2330 Joules of energy. What is the specific heat of the metal? 3. A hot 1 kg chunk of copper is allowed to cool to 100°C. If the copper gave off 231 kJ of energy, what was the initial temperature of the copper? The specific heat of copper is 0.385 J/g°C.

This solution involves calculating energy required to heat lead, finding the specific heat of a metal ball, and determining the initial temperature of a cooling copper chunk using the formula Q = mcΔT. Suitable for students in Grades 9-12.

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