Question

Francesca is meeting Mercedes at a store. The line labeled F(m) shows the path Francesca will take, and the line labeled M (m) represents the path of Mercedes. How long will it take for them to arrive at the store?

Answer 1

I don't have the graph or the equations, so I can't give a single numeric answer. I can, however, tell you exactly how to get the time from the picture (or solve it if you paste the graph points or the two equations).

Steps to find how long until they arrive (i.e. when Francesca and Mercedes reach the same place at the same time):

1. Confirm the axes: the horizontal axis should be time (t, in minutes or seconds) and the vertical axis distance from some reference (meters).
2. From the graph read two clear points on Francesca's line and two on Mercedes's line. Use those to get each line in slope–intercept form:
- slope (speed) = (change in distance)/(change in time) = m
- line: distance(t) = m·t + b
3. You will then have F(t) = m_F·t + b_F and M(t) = m_M·t + b_M.
4. Solve F(t) = M(t): (m_F − m_M)·t = b_M − b_F, so
t = (b_M − b_F) / (m_F − m_M).
- If m_F = m_M and b_F ≠ b_M, they never meet. If m_F = m_M and b_F = b_M, they are together at all times shown.
5. Check units of t (minutes, seconds) and that the intersection time is within the time domain shown on the graph (nonnegative and within the plotted range).

If you post the graph (or give two points from each line or the two line equations), I’ll compute the numerical value for you.