1. **State the problem:** We have a right-angled triangle ABC with angle ABC = 90°. AB = 5.3 cm (base, horizontal), BC = 4.8 cm (vertical), both correct to 2 significant figures. We need to calculate the lower bound for the gradient of the line AC. 2. **Recall the gradient formula:** The gradient (slope) of line AC is given by $$\text{gradient} = \frac{\text{vertical change}}{\text{horizontal change}} = \frac{\text{height}}{\text{base}}$$ Here, vertical change = BC, horizontal change = AB. 3. **Understand bounds:** Since AB and BC are given to 2 significant figures, their bounds are: - For AB = 5.3 cm, the lower bound is $5.25$ cm and the upper bound is $5.35$ cm. - For BC = 4.8 cm, the lower bound is $4.75$ cm and the upper bound is $4.85$ cm. 4. **Calculate the lower bound for the gradient:** The gradient is $\frac{BC}{AB}$. To get the lower bound of the gradient, we want the smallest possible value of $\frac{BC}{AB}$. Since gradient increases with BC and decreases with AB, the lower bound is when BC is at its lower bound and AB is at its upper bound. 5. **Substitute the bounds:** $$\text{lower bound gradient} = \frac{4.75}{5.35}$$ 6. **Simplify the fraction:** $$\frac{4.75}{5.35} = \frac{\cancel{4.75}}{\cancel{5.35}}$$ (We show cancellation but here no common factor to cancel exactly, so we proceed to decimal division.) 7. **Calculate the decimal value:** $$\frac{4.75}{5.35} \approx 0.8879$$ 8. **Final answer:** The lower bound for the gradient of line AC is approximately $$\boxed{0.89}$$