1. Use the first law of thermodynamics in rate form:

$\frac{dQ}{dt} = \frac{dU}{dt} + \frac{dW}{dt}$

where:

• $\frac{dQ}{dt}$ = rate of heat supplied to the system
• $\frac{dU}{dt}$ = rate of increase of internal energy
• $\frac{dW}{dt}$ = rate at which work is done by the system

2. Given:

$\frac{dQ}{dt} = 1000\ \text{W}$ $\frac{dW}{dt} = 200\ \text{W}$

3. Substitute into the first law:

$1000 = \frac{dU}{dt} + 200$

4. Solve for $\frac{dU}{dt}$:

$\frac{dU}{dt} = 1000 - 200 = 800\ \text{W}$

5. Therefore, the rate at which the internal energy increases is:

$\boxed{800\ \text{W}}$

6. Option check:

• A: $600\ \text{W}$ — incorrect
• B: $1200\ \text{W}$ — incorrect
• C: $500\ \text{W}$ — incorrect
• D: $800\ \text{W}$ — correct