Answer
$T=666.7N$
Work Step by Step
The formula for average power states that $$\overline P=\frac{W}{t}$$ $$W=\overline Pt$$
The difference between 2 cases is that when there is a skier behind, the engine must do more work to pull the skier, which is translated in the tension $T$ in the rope. In other words,
- When there is no skier: $W_{other forces}=(7.5\times10^4)t$
- When there is a skier: $W_T+W_{other forces}=(8.3\times10^4)t$
From these 2 equations, we can see that $$W_T=(8.3\times10^4)t-(7.5\times10^4)t=(0.8\times10^4)t (1)$$
Next, we know $W_T=(T\cos\theta)s$
$T$ acts as the force pulling the skier along the movement, so $\cos\theta=\cos0=1$
The distance traveled $s=v_{avg}t=12t$
So, $W_T=12Tt$
Plug this back to (1), we can calculate $T$: $$T=\frac{0.8\times10^4}{12}=666.7N$$
