1. **State the problem:** We have data of time to climb (in minutes) and height of rock formations (in feet). We want to find the line of best fit to model the relationship between time and height. 2. **Formula for line of best fit:** The line of best fit is generally given by the equation $$y = mx + b$$ where $y$ is the dependent variable (height), $x$ is the independent variable (time), $m$ is the slope, and $b$ is the y-intercept. 3. **Important rules:** The slope $m$ is calculated by $$m = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2}$$ and the intercept $b$ by $$b = \frac{\sum y - m \sum x}{n}$$ where $n$ is the number of data points. 4. **Calculate sums:** Let $x$ be time and $y$ be height. $$\sum x = 30 + 45 + 115 + 87 + 160 + 60 + 135 + 117 + 55 + 145 + 95 + 150 + 67 + 120 + 100 + 88 + 138 + 75 + 167 + 120 = 1974$$ $$\sum y = 85 + 120 + 298 + 194 + 434 + 241 + 351 + 339 + 180 + 400 + 248 + 370 + 155 + 296 + 271 + 255 + 315 + 215 + 405 + 317 = 5989$$ Calculate $$\sum xy$$ and $$\sum x^2$$: $$\sum xy = 30\times85 + 45\times120 + 115\times298 + 87\times194 + 160\times434 + 60\times241 + 135\times351 + 117\times339 + 55\times180 + 145\times400 + 95\times248 + 150\times370 + 67\times155 + 120\times296 + 100\times271 + 88\times255 + 138\times315 + 75\times215 + 167\times405 + 120\times317 = 703,964$$ $$\sum x^2 = 30^2 + 45^2 + 115^2 + 87^2 + 160^2 + 60^2 + 135^2 + 117^2 + 55^2 + 145^2 + 95^2 + 150^2 + 67^2 + 120^2 + 100^2 + 88^2 + 138^2 + 75^2 + 167^2 + 120^2 = 217,974$$ 5. **Calculate slope $m$:** $$m = \frac{20 \times 703,964 - 1974 \times 5989}{20 \times 217,974 - 1974^2} = \frac{14,079,280 - 11,819,886}{4,359,480 - 3,896,676} = \frac{2,259,394}{462,804} \approx 4.88$$ 6. **Calculate intercept $b$:** $$b = \frac{5989 - 4.88 \times 1974}{20} = \frac{5989 - 9629.12}{20} = \frac{-3640.12}{20} = -182.01$$ 7. **Final line of best fit:** $$y = 4.88x - 182.01$$ This means for each additional minute of climbing time, the height increases by approximately 4.88 feet, starting from an intercept of -182.01 feet (which is an extrapolation and may not be physically meaningful).