1. **State the problem:** We know that the distance $d$ a marble falls is directly proportional to the square of the time $t$ it has been falling. Given that it falls 42.3 metres in 3 seconds, we want to find how far it falls in 5 seconds. 2. **Write the formula for direct proportionality:** Since $d$ is directly proportional to $t^2$, we write: $$ d = k t^2 $$ where $k$ is the constant of proportionality. 3. **Find the constant $k$ using the given data:** Given $d = 42.3$ metres when $t = 3$ seconds, substitute these values: $$ 42.3 = k \times 3^2 $$ $$ 42.3 = 9k $$ Solve for $k$: $$ k = \frac{42.3}{9} = 4.7 $$ 4. **Use $k$ to find the distance fallen in 5 seconds:** Substitute $t = 5$ and $k = 4.7$ into the formula: $$ d = 4.7 \times 5^2 = 4.7 \times 25 = 117.5 $$ 5. **Final answer:** The marble falls **117.5 metres** in 5 seconds. This method uses the rule of direct proportionality and substitution to find the unknown distance.