1. The problem states there is a proportional relationship between the number of days Tanvi is stranded on the island, $x$, and the total number of trees she cuts down, $y$. 2. A proportional relationship means $y = kx$, where $k$ is the constant of proportionality. 3. From the data, when $x = 8$, $y = 40$. 4. Substitute these values into the formula to find $k$: $$40 = k \times 8$$ 5. Solve for $k$: $$k = \frac{40}{8}$$ 6. Simplify the fraction: $$k = \frac{\cancel{40}}{\cancel{8}} = 5$$ 7. Therefore, the constant of proportionality is $5$. This means Tanvi cuts down 5 trees per day on the island.