A jogger's internal energy changes because he performs `6.4 xx 10^5 ` J of work and gives off `4.9 xx 10^5` of heat. However, to cause the same change in his internal energy while walking, he must do `8.2 xx 10^5 ` J of work. Determine the magnitude of the heat given off while walking.

Options

A. `3.6 xx 10^5 J`
B. `4.0 xx 10^5 J`
C. `2.4 xx 10^5 J`
D. `3.1 xx 10^5 J`

Answer

To solve the problem, we will apply the First Law of Thermodynamics, which states:

\[ \Delta U = Q + W \]

Where:
- \(\Delta U\) is the change in internal energy,
- \(Q\) is the heat added to the system (positive if added, negative if given off),
- \(W\) is the work done on the system (positive if done on the system, negative if done by the system).

### Step 1: Determine the change in internal energy for jogging

For the jogger, we have the following values:
- Work done while jogging, \(W_1 = 6.4 \times 10^5 \, \text{J}\) (positive because work is done on the system),
- Heat given off while jogging, \(Q_1 = -4.9 \times 10^5 \, \text{J}\) (negative because heat is leaving the system).

Using the First Law of Thermodynamics:

\[ \Delta U = Q_1 + W_1 \]

Substituting the values:

\[ \Delta U = (-4.9 \times 10^5) + (6.4 \times 10^5) \]

Calculating:

\[ \Delta U = 1.5 \times 10^5 \, \text{J} \]

### Step 2: Determine the heat given off while walking

Now, for walking, we know:
- Work done while walking, \(W_2 = 8.2 \times 10^5 \, \text{J}\) (positive),
- Change in internal energy is the same as jogging, \(\Delta U = 1.5 \times 10^5 \, \text{J}\).

We need to find \(Q_2\) (the heat given off while walking). Again, using the First Law of Thermodynamics:

\[ \Delta U = Q_2 + W_2 \]

Rearranging for \(Q_2\):

\[ Q_2 = \Delta U - W_2 \]

Substituting the known values:

\[ Q_2 = (1.5 \times 10^5) - (8.2 \times 10^5) \]

Calculating:

\[ Q_2 = 1.5 \times 10^5 - 8.2 \times 10^5 = -6.7 \times 10^5 \, \text{J} \]

### Step 3: Determine the magnitude of heat given off

Since we are asked for the magnitude of heat given off, we take the absolute value:

\[ |Q_2| = 6.7 \times 10^5 \, \text{J} \]

### Final Answer
The magnitude of the heat given off while walking is:

\[ \boxed{6.7 \times 10^5 \, \text{J}} \]