Question

The grade of copper (in wt%) of an ore body determined at locations 1, 2 and 3 are indicated (in parentheses) below. The grade of copper at an unknown location x calculated using Inverse-Square Distance Weighting (IDW) is \(\underline{\hspace{1cm}}\) wt %. (round off to 2 decimal places) 

Hint:

Inverse-Square Distance Weighting (IDW) gives more weight to points closer to the target location. Use the formula to calculate the grade at the unknown location.

Answer 1

Correct Answer: 1.67 - 1.69

Inverse-Square Distance Weighting (IDW) is a spatial interpolation method where the weight is inversely proportional to the square of the distance between the known data points and the location to be estimated. The formula for IDW is: \[ \text{Grade at x} = \frac{\sum \left( \frac{G_i}{d_i^2} \right)}{\sum \left( \frac{1}{d_i^2} \right)} \] where: - \( G_i \) is the grade at location \( i \), - \( d_i \) is the distance between location \( i \) and \( x \). Given:
- Location 1: \( G_1 = 1.5% \), \( d_1 = 2 \, \text{m} \),
- Location 2: \( G_2 = 1.8% \), \( d_2 = 3 \, \text{m} \),
- Location 3: \( G_3 = 2.5% \), \( d_3 = 5 \, \text{m} \).
First, calculate the weights: \[ w_1 = \frac{1}{d_1^2} = \frac{1}{2^2} = 0.25, w_2 = \frac{1}{d_2^2} = \frac{1}{3^2} = 0.1111, w_3 = \frac{1}{d_3^2} = \frac{1}{5^2} = 0.04. \] Now, calculate the weighted sum: \[ \text{Grade at x} = \frac{(1.5 \times 0.25) + (1.8 \times 0.1111) + (2.5 \times 0.04)}{0.25 + 0.1111 + 0.04} \approx \frac{(0.375) + (0.2) + (0.1)}{0.4011} = \frac{0.675}{0.4011} \approx 1.68. \] Thus, the grade of copper at location x is \( \boxed{1.67} \, % \).