Question

on the first day after the new moon, 2% of the moon’s surface is illuminated. on the second day, 6% of the moon’s surface is illuminated. assuming this data can be modeled with a linear function, complete the table. round to the nearest day if necessary. day number illumination 1 2% 2 6%... 50% 100%

Explanation:

Step1: Find the slope of the linear function

The two points are \((1, 2\%)\) and \((2, 6\%)\). The slope \(m\) is calculated as \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{6 - 2}{2 - 1}=4\). So the linear equation is \(y - y_1=m(x - x_1)\), substituting \(x_1 = 1\), \(y_1 = 2\), we get \(y=4(x - 1)+2=4x - 4 + 2=4x - 2\).

Step2: Find the day for 50% illumination

Set \(y = 50\) in the equation \(y = 4x - 2\). Then \(50=4x - 2\). Add 2 to both sides: \(50 + 2=4x\), so \(52 = 4x\). Divide both sides by 4: \(x=\frac{52}{4}=13\).

Step3: Find the day for 100% illumination

Set \(y = 100\) in the equation \(y = 4x - 2\). Then \(100=4x - 2\). Add 2 to both sides: \(100 + 2=4x\), so \(102 = 4x\). Divide both sides by 4: \(x=\frac{102}{4}=25.5\approx26\) (rounded to the nearest day).

Answer:

For 50% illumination, the day number is 13. For 100% illumination, the day number is 26.

So the completed table (for the two missing rows) is:

Day Number	Illumination
26	100%