Using a total station, the rectangular dimensions of a building are measured, it is 600.870 ± 0.019 m by 350.080 ± 0.016 m. Assuming only errors in the remote observations, calculate: • Area delimited by the building and its error propagation • Building perimeter and its error propagation

Let's start by calculating the area of the building:

The area, A, of a rectangle is given by:
A = l \times w
where l is the length and w is the width of the rectangle.

Given length, l = 600.870 \pm 0.019 m and width, w = 350.080 \pm 0.016 m.

To find the area:
A=600.870\times350.080=210352.570\,m^2

Now, let's calculate the error in the area:
The formula for error propagation when multiplying two quantities is given by:
\sigma_{A} = \sqrt{(w \times \sigma_{l})^2 + (l \times \sigma_{w})^2}
where \sigma_{A} is the error in the area, \sigma_{l} is the error in the length, \sigma_{w} is the error in the width.

Calculating the error in the area:
\sigma_A=\sqrt{(350.080 \times0.019)^2 + (600.870 \times0.016)^2}=\sqrt{44.2427+92.4274}=\sqrt{136.6701}\approx11.691\,m^2

Therefore, the area of the building is 210352.570\,m^2 with an error of approximately 11.691\,m^2 .

Now, let's calculate the perimeter of the building:

The perimeter, P, of a rectangle is given by:
P = 2(l + w)

Given length, l = 600.870 \pm 0.019 m and width, w = 350.080 \pm 0.016 m.

To find the perimeter:
P=2(600.870+350.080)=1901.900\,m

Now, let's calculate the error in the perimeter:
The formula for error propagation when summing two quantities is given by:
\sigma_{P} = \sqrt{\sigma_{l}^2 + \sigma_{w}^2}
where \sigma_{P} is the error in the perimeter, \sigma_{l} is the error in the length, \sigma_{w} is the error in the width.

Calculating the error in the perimeter:
\sigma_{P} = \sqrt{0.019^2 + 0.016^2} = \sqrt{0.000361 + 0.000256} = \sqrt{0.000617} \approx 0.025 \,m

Therefore, the perimeter of the building is 1901.9 \,m with an error of approximately 0.025 \,m .

\textbf{Answer:}
1. The area of the building is 210352.570 m² with an error of approximately 11.691 m².
2. The perimeter of the building is 1901.9 m with an error of approximately 0.025 m.