Let's solve the questions one by one:

(a) Lower Bound for the Lengths of $AB$ and $BC$

1. (i) Lower bound for $AB$:

   The length $AB = 20$ cm, correct to 1 significant figure. The lower bound is calculated as: $\text{Lower bound} = 20 - 0.5 = 19.5 \text{ cm}$

2. (ii) Lower bound for $BC$:

   The length $BC = 8.3$ cm, correct to 2 significant figures. The lower bound is calculated as: $\text{Lower bound} = 8.3 - 0.05 = 8.25 \text{ cm}$

(b) Lower Bound for the Area of Triangle $ABC$

The area of a right-angled triangle is given by: $\text{Area} = \frac{1}{2} \times AB \times BC$ Using the lower bounds: $\text{Lower bound of Area} = \frac{1}{2} \times 19.5 \times 8.25 = 80.4375 \text{ cm}^2$

(c) Lower Bound for the Value of $\tan x^\circ$

The tangent of angle $x^\circ$ is given by: $\tan x^\circ = \frac{BC}{AB}$ Using the lower bounds: $\text{Lower bound of } \tan x^\circ = \frac{8.25}{20.5} \approx 0.4024$

These are the calculated lower bounds. Let me know if you would like to see more details or have any questions about this solution!

5 Relative Questions:

1. What is the upper bound for the length of $AB$ and $BC$?
2. How does changing the number of significant figures affect the lower bound calculation?
3. How would you find the upper bound of the area of triangle $ABC$?
4. How can we verify the calculated lower bounds using the concept of error intervals?
5. How would the lower bound of $\tan x^\circ$ change if $AB$ had 2 significant figures?

Tip:

When dealing with measurements and bounds, always keep track of the precision and significant figures to ensure accurate calculations.