Let's solve the problem step by step: ## **Step 1: Arrange the Data** Let \( x \) = RNA (μg) Let \( y \) = Absorbance at 260 nm | \( x \) (μg) | \( y \) (Abs) | |--------------|--------------| | 0.00 | 0.160 | | 10.89 | 0.472 | | 21.78 | 0.714 | | 32.67 | 1.032 | | 43.56 | 1.258 | --- ## **Step 2: Least Squares Method Formula** The equation of the best fit line is: \[ y = mx + b \] where \( m \) is the slope and \( b \) is the intercept. ### **Formulas:** \[ m = \frac{N\sum(xy) - \sum x \sum y}{N\sum(x^2) - (\sum x)^2} \] \[ b = \frac{\sum y - m\sum x}{N} \] Where \( N \) is the number of data points (here, \( N = 5 \)). --- ## **Step 3: Calculate the Sums** | \( x \) | \( y \) | \( x^2 \) | \( y^2 \) | \( x y \) | |-----------|-----------|---------------|---------------|----------------| | 0.00 | 0.160 | 0.000 | 0.0256 | 0.000 | | 10.89 | 0.472 | 118.6081 | 0.222784 | 5.13768 | | 21.78 | 0.714 | 474.7684 | 0.509796 | 15.54372 | | 32.67 | 1.032 | 1067.3289 | 1.065024 | 33.74944 | | 43.56 | 1.258 | 1897.0836 | 1.582564 | 54.78248 | \[ \sum x = 0 + 10.89 + 21.78 + 32.67 + 43.56 = 108.90 \] \[ \sum y = 0.160 + 0.472 + 0.714 + 1.032 + 1.258 = 3.636 \] \[ \sum x^2 = 0 + 118.6081 + 474.7684 + 1067.3289 + 1897.0836 = 3557.789 \] \[ \sum xy = 0 + 5.13768 + 15.54372 + 33.74944 + 54.78248 = 109.2133 \] --- ## **Step 4: Plug into the Formulas** \[ m = \frac{5 \times 109.2133 - 108.90 \times 3.636}{5 \times 3557.789 - (108.90)^2} \] Calculate numerator: \[ 5 \times 109.2133 = 546.0665 \] \[ 108.90 \times 3.636 = 395.7384 \] \[ \text{Num} = 546.0665 - 395.7384 = 150.3281 \] Calculate denominator: \[ 5 \times 3557.789 = 17788.945 \] \[ (108.90)^2 = 11862.81 \] \[ \text{Den} = 17788.945 - 11862.81 = 5926.135 \] So, \[ m = \frac{150.3281}{5926.135} \approx 0.0254 \ \mu g^{-1} \] \[ b = \frac{3.636 - (0.0254 \times 108.90)}{5} \] \[ 0.0254 \times 108.90 = 2.764 \] \[ 3.636 - 2.764 = 0.872 \] \[ b = \frac{0.872}{5} = 0.174 \] --- ## **Step 5: Find Mass for Absorbance 0.620** The equation is \( y = mx + b \). Given \( y = 0.620 \): \[ 0.620 = 0.0254x + 0.174 \] \[ 0.620 - 0.174 = 0.0254x \] \[ 0.446 = 0.0254x \] \[ x = \frac{0.446}{0.0254} \approx 17.56\ \mu g \] --- ## **Final Answers** \[ \boxed{m = 0.0254\ \mu g^{-1}} \] \[ \boxed{b = 0.174} \] \[ \text{mass} = \boxed{17.56\ \mu g} \]