1. **Problem statement:** We are given that on day 1, 2% of the moon's surface is illuminated, and on day 2, 6% is illuminated. We need to predict the day when the illumination reaches 50% and 100%, check if the 100% illumination on day 14 agrees with the prediction, and determine if the illumination is a linear function of the day. 2. **Step 1: Define the linear function** We assume the percentage illumination $y$ is a linear function of the day $x$, so: $$y = mx + b$$ where $m$ is the slope and $b$ is the y-intercept. 3. **Step 2: Use given points to find $m$ and $b$** Given points: $(1, 2)$ and $(2, 6)$. Calculate slope: $$m = \frac{6 - 2}{2 - 1} = \frac{4}{1} = 4$$ Use point-slope form with point $(1, 2)$: $$2 = 4 \times 1 + b \implies b = 2 - 4 = -2$$ So the function is: $$y = 4x - 2$$ 4. **Step 3: Predict day for 50% illumination** Set $y = 50$: $$50 = 4x - 2$$ Add 2 to both sides: $$50 + 2 = 4x$$ $$52 = 4x$$ Divide both sides by 4: $$x = \frac{\cancel{52}}{\cancel{4}} = 13$$ So, 50% illumination is predicted on day 13. 5. **Step 4: Predict day for 100% illumination** Set $y = 100$: $$100 = 4x - 2$$ Add 2: $$102 = 4x$$ Divide by 4: $$x = \frac{\cancel{102}}{\cancel{4}} = 25.5$$ So, 100% illumination is predicted on day 25.5. 6. **Step 5: Compare prediction with actual data** The moon’s surface is 100% illuminated on day 14, but our prediction was day 25.5. This does not agree with the prediction. 7. **Step 6: Is the illumination a linear function?** Since the actual 100% illumination day (14) differs significantly from the predicted day (25.5), the illumination percentage is not a linear function of the day. **Final answers:** - a) 50% illumination on day 13, 100% illumination on day 25.5 (predicted) - b) No, the actual 100% illumination on day 14 does not agree with the prediction. - c) No, the percentage illumination is not a linear function of the day.