Question

On an old plan of scale 1:1000, leasehold area of a mine is now measured as 802 cm^2 using a planimeter. The plan is found to have shrunk, such that the original line of 10 cm is now measured as 9.8 cm on the plan. True lease hold mine area, in m^2 is \(\underline{\hspace{2cm}}\) (round off to the nearest integer).

Hint:

When dealing with scale drawings, adjust areas by squaring the ratio of the lengths to get the true area.

Answer 1

Correct Answer: 83450

Given: \[ \text{Original length} = 10 \, \text{cm}, \text{Measured length} = 9.8 \, \text{cm}, \text{Area on plan} = 802 \, \text{cm}^2 \] The scale of the plan is 1:1000, meaning that 1 cm on the plan represents 1000 cm in real life. Therefore, the true area in real life is calculated by adjusting for the scale. The ratio of the original length to the measured length is: \[ \frac{\text{Original length}}{\text{Measured length}} = \frac{10}{9.8} = 1.02041 \] The true area is obtained by multiplying the area on the plan by the square of the ratio of the lengths: \[ \text{True area} = 802 \times \left( \frac{10}{9.8} \right)^2 = 802 \times 1.041 \approx 834.50 \, \text{m}^2 \] Thus, the true leasehold mine area is approximately: \[ \boxed{834.50 \, \text{m}^2} \]