Here is a solution to each question in the image provided.

Question 1:

The height of horses is often measured in "hands." The reason this may be a poor standard of length is due to the variability and lack of consistency in hand size across different individuals, leading to unreliable and non-standard measurements.

Correct Answer: (c) The width of the hand will vary as finger nails grow and get cut.

Question 2:

For each calculation, round the final result to the appropriate level of accuracy based on the given significant figures.

1. (i) $256 + 28.2 + 0.75 + 9.6$

   • Result: 294.55, rounded to 295 (3 significant figures).

2. (ii) $6.280 \times \pi$

   • Result: $6.280 \times 3.14159 \approx 19.7372$, rounded to 19.7 (3 significant figures).

3. (iii) $0.0018 \times 712.8$

   • Result: $0.0018 \times 712.8 = 1.28256$, rounded to 1.3 (2 significant figures).

4. (iv) $\frac{1.98 \times 10^4}{2.71 \times \sqrt{8.112}}$

   • First, calculate $\sqrt{8.112} \approx 2.85$.
   • Then, $2.71 \times 2.85 = 7.7235$.
   • Finally, $\frac{1.98 \times 10^4}{7.7235} \approx 2563.2$, rounded to 2560 (3 significant figures).

Question 3:

To find the frequency $f$ of a sound wave with wavelength $\lambda = 4.0 \, \text{m}$ and speed of sound $c = 3.0 \times 10^8 \, \text{m/s}$: $f = \frac{c}{\lambda} = \frac{3.0 \times 10^8}{4.0} = 7.5 \times 10^7 \, \text{Hz}$

Question 4:

In the expression $x = C t^2$, the units of $C$ can be derived by dimensional analysis. Since $x$ has units of meters (m) and $t$ has units of seconds (s): $\text{Units of } C = \frac{\text{m}}{\text{s}^2} = \text{m/s}^2$

Question 5:

For the expression $x = E \cos(2\pi f t)$, the argument of the cosine function must be dimensionless. Therefore, $E$ has the same units as $x$, which are meters (m).

Question 6:

Convert the units as follows:

1. (i) $343 \, \text{m/s}$ to km/h: $343 \, \text{m/s} = 343 \times 3.6 = 1234.8 \, \text{km/h}$

2. (ii) $998 \, \text{mm}$ to m: $998 \, \text{mm} = 0.998 \, \text{m}$

3. (iii) $998 \, \text{mm}^2$ to $\text{m}^2$: $998 \, \text{mm}^2 = 998 \times 10^{-6} = 0.000998 \, \text{m}^2$

4. (iv) $8940 \, \text{kg/m}^3$ to $\text{g/cm}^3$: $8940 \, \text{kg/m}^3 = 8.94 \, \text{g/cm}^3$

Would you like detailed explanations for each calculation?

Additional Questions for Practice:

1. What would be an example where "hands" as a unit is a practical measurement?
2. How do you determine the correct number of significant figures in a calculation?
3. What is the relationship between frequency, wavelength, and wave speed?
4. How does dimensional analysis help in identifying the units of constants?
5. Why must the argument of a trigonometric function be dimensionless?

Tip:

When working with unit conversions, always ensure your answer has the same level of precision as the input measurements.