A surveyor needs to compute the area of two different land parcels:

Recommend the most appropriate mathematical or mechanical method for calculating the area of each parcel and explain why.

A land developer is using a historical plat map to estimate the area of a large parcel. The map is drawn at a scale of 1:5000. They use a planimeter to trace the boundary and calculate an area of $4000 \text{ cm}^2$ on the paper. However, they accidentally use the wrong conversion factor, assuming the scale was 1:1000.

By what factor will their calculated ground area be incorrect? Explain the mathematical relationship between linear scale and area scale.

Calculate the area of a closed traverse defined by the following adjusted coordinates (in meters):

A traverse ABCD has the following adjusted latitudes and departures:

Calculate the area using the Double Meridian Distance (DMD) method.

Offsets from a straight baseline to an irregular boundary are measured at regular intervals of $10 \text{ m}$. The measured offsets (in meters) are: $h_1 = 3.5$, $h_2 = 4.2$, $h_3 = 5.0$, $h_4 = 4.8$, $h_5 = 3.9$, $h_6 = 2.5$, $h_7 = 0.0$ (boundary meets baseline).

Calculate the area bounded by the baseline, the offsets, and the irregular boundary using the Trapezoidal Rule.

Using the exact same offset data from Problem 3 ($d = 10 \text{ m}$; offsets: $3.5, 4.2, 5.0, 4.8, 3.9, 2.5, 0.0$), calculate the area using Simpson's 1/3 Rule.