1. **Stating the problem:** We are given a table showing the relationship between burette readings and volumes in a measuring cylinder for various numbers of drops. We want to plot a graph of burette reading (cm³) versus measuring cylinder reading (cm³) and analyze it. 2. **Data points from the table:** - For 25 drops: (1.50, 1.40) - For 50 drops: (2.50, 2.60) - For 75 drops: (4.50, 4.00) - For 100 drops: (5.50, 5.40) - For 125 drops: (6.50, 6.80) - For 150 drops: (8.00, 7.00) - For 200 drops: (9.50, 9.40) 3. **Plotting the graph:** Plot burette reading $x$ (cm³) on the horizontal axis and measuring cylinder reading $y$ (cm³) on the vertical axis using the above points. 4. **Finding the best straight line (least squares fit):** Use the points to estimate a linear relation: $y = mx + c$. Calculate the slope $m$ and intercept $c$: Calculate means: $$ \bar{x} = \frac{1.50 + 2.50 + 4.50 + 5.50 + 6.50 + 8.00 + 9.50}{7} = \frac{38}{7} \approx 5.43 $$ $$ \bar{y} = \frac{1.40 + 2.60 + 4.00 + 5.40 + 6.80 + 7.00 + 9.40}{7} = \frac{36.60}{7} \approx 5.23 $$ Calculate slope $m$: $$ m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} $$ Calculate numerator: $$(1.50 - 5.43)(1.40 - 5.23) + (2.50 - 5.43)(2.60 - 5.23) + (4.50 - 5.43)(4.00 - 5.23) + (5.50 - 5.43)(5.40 - 5.23) + (6.50 - 5.43)(6.80 - 5.23) + (8.00 - 5.43)(7.00 - 5.23) + (9.50 - 5.43)(9.40 - 5.23) $$ $$ = (-3.93)(-3.83) + (-2.93)(-2.63) + (-0.93)(-1.23) + (0.07)(0.17) + (1.07)(1.57) + (2.57)(1.77) + (4.07)(4.17) $$ $$ = 15.05 + 7.71 + 1.14 + 0.01 + 1.68 + 4.55 + 16.97 = 46.11 $$ Calculate denominator: $$(1.50 - 5.43)^2 + (2.50 - 5.43)^2 + (4.50 - 5.43)^2 + (5.50 - 5.43)^2 + (6.50 - 5.43)^2 + (8.00 - 5.43)^2 + (9.50 - 5.43)^2 $$ $$ = (-3.93)^2 + (-2.93)^2 + (-0.93)^2 + (0.07)^2 + (1.07)^2 + (2.57)^2 + (4.07)^2 $$ $$ = 15.45 + 8.58 + 0.86 + 0.005 + 1.14 + 6.60 + 16.57 = 49.20 $$ So, $$ m = \frac{46.11}{49.20} \approx 0.937 $$ Calculate intercept $c$: $$ c = \bar{y} - m \bar{x} = 5.23 - 0.937 \times 5.43 = 5.23 - 5.09 = 0.14 $$ Thus the best-fit line is: $$ y = 0.937 x + 0.14 $$ 5. **Use the graph to determine (i) the measuring cylinder reading for burette volume 4.0 cm³:** Using the equation: $$ y = 0.937 \times 4.0 + 0.14 = 3.75 + 0.14 = 3.89$$ 6. **Use the graph to determine (ii) the burette reading for measuring cylinder reading 8.2 cm³:** Rearranged: $$ x = \frac{y - c}{m} = \frac{8.2 - 0.14}{0.937} = \frac{8.06}{0.937} \approx 8.60 $$ 7. **Compare answers and accounts:** - (i) The measuring cylinder reading for 4.0 cm³ from the burette is about 3.89 cm³, slightly less due to possible losses or experimental error. - (ii) The burette volume corresponding to 8.2 cm³ in the measuring cylinder is approximately 8.60 cm³, indicating slight discrepancy possibly due to overflow, evaporation, or calibration differences. These small differences are normal in practical lab settings.