Answer :
To address the problem, we need to work through it step-by-step carefully:
(i) Shrunk Scale
The original scale was 10 meters to 1 centimeter (10 m : 1 cm), meaning that 1 cm on the plan represents 10 meters in reality. However, the scale has changed because the plan has shrunk.
The original line length was 10 cm, and it shrunk to 9.7 cm. Thus, the scale also changes.
To find the new scale (Shrunk Scale):
Since the line shrunk by a factor of [tex]\frac{9.7}{10} = 0.97[/tex], the new scale is:
[tex]\text{Shrunk Scale} = 10 m : (1 \text{ cm} \times 0.97) = 10 m : 0.97 \text{ cm}[/tex]
(ii) True Length of Line if Measured Length is 14.75 cm
Using the shrunk scale (where 0.97 cm represents 10 meters), to find the true length:
[tex]\text{True Length} = \frac{14.75 \text{ cm} \times 10 \text{ m/cm}}{0.97 \text{ cm}} = \frac{147.5 \text{ m}}{0.97} \approx 152.06 \text{ m}[/tex]
So, the true length of the line is approximately 152.06 meters.
(iii) True Area of Plot if the Measured Area is 100.2 cm²
The area changes by the square of the scale factor since area is two-dimensional.
The scale factor for length is [tex]\frac{9.7}{10} = 0.97[/tex].
Thus, the scale factor for the area will be [tex](0.97)^2 = 0.9409[/tex].
Given the measured area is 100.2 cm², the true area is:
[tex]\text{True Area} = \frac{100.2 \text{ cm}^2}{0.9409} \approx 106.46 \text{ cm}^2[/tex]
Converting this back to square meters using the original scale where 1 cm represents 10 meters, hence 1 cm² represents 100 m²:
[tex]\text{True Area in m}^2 = 106.46 \times 100 \approx 10646 \text{ m}^2[/tex]
Thus, the true area of the plot is approximately 10,646 square meters.
