Final answer:
The distance between the two parallel lines y=1/3x+5 and y=1/3x+15 is 3 units, calculated using the formula for the distance between parallel lines with a shared slope and differing y-intercepts.
Step-by-step explanation:
The student has asked to find the distance between two parallel lines, given by the equations y=\frac{1}{3}x+5 and y=\frac{1}{3}x+15. To determine the distance between these two lines, we recognize that the lines have the same slope but different y-intercepts, which means they are parallel and never intersect.
The distance between two parallel lines in the slope-intercept form, y=mx+b, can be found using the formula d=\fracb_2-b_1{\sqrt{1+m^2}}, where b_1 and b_2 are the y-intercepts of the two lines and m is the slope of the lines.
Plugging the values b_1=5, b_2=15, and m=\frac{1}{3} into the formula, we get:
d=\frac15-5{\sqrt{1+\left(\frac{1}{3}\right)^2}} = \frac{10}{\sqrt{1+\frac{1}{9}}} = \frac{10}{\sqrt{\frac{10}{9}}} = \frac{10}{\frac{10}{3}} = 3
The distance between the two lines y = 1/3x + 5 and y = 1/3x + 15 can be found by subtracting the y-intercepts of both lines. In this case, the y-intercepts are 5 and 15 respectively. So, the distance between the lines is 15 - 5 = 10 units.
Therefore, the distance between the two lines is 3 units.