Express the Jaguar's Final Speed: Calculate X in Terms of Average Speed

To solve this problem, let's outline the detailed steps:

• Step 1: Calculate the total time

The jaguar starts the chase after $X$ minutes of stalking and then chases for 8 minutes. For the last part, the chase lasts 4 more minutes. So, the total time in minutes is $X + 8 + 4 = X + 12$. To convert this to hours, divide by 60, resulting in $\frac{X + 12}{60}$ hours.

• Step 2: Use the average speed to find the total distance

The average speed over the total time is 80 km/h. Therefore, the total distance $D_{\text{total}}$ is:

$D_{\text{total}} = 80 \times \left(\frac{X + 12}{60}\right)$

$D_{\text{total}} = \frac{80(X + 12)}{60}$

$D_{\text{total}} = \frac{4(X + 12)}{3}$

• Step 3: Calculate the distance for the first two segments

For the second segment, the jaguar's speed is 70 km/h for 8 minutes. Convert 8 minutes to hours to find the distance:

$D_{\text{second}} = 70 \times \left(\frac{8}{60}\right) = \frac{70}{60} \times 8 = \frac{560}{60} = \frac{28}{3}$

• Step 4: Find the distance for the final segment and speed

The distance in the last segment $D_{\text{third}}$ is the total distance minus the distance covered in the first two parts:

$D_{\text{third}} = \frac{4(X + 12)}{3} - \frac{28}{3}$

$D_{\text{third}} = \frac{4X + 48 - 28}{3} = \frac{4X + 20}{3}$

The time for the last segment is 4 minutes or $\frac{4}{60}$ hours. Let the speed during this segment be $v_{\text{third}}$. Thus:

$v_{\text{third}} \times \frac{4}{60} = \frac{4X + 20}{3}$

$v_{\text{third}} = \frac{4X + 20}{3} \times \frac{60}{4}$

$v_{\text{third}} = \frac{(4X + 20) \times 15}{3}$

$v_{\text{third}} = \frac{60X + 300}{3} = 20X + 100$

The final speed of the jaguar during the last 4 minutes, in terms of $X$, is $\boldsymbol{100 + 20X}$ km/h.