Calculus word problem

John wishes to time his approach to the bus stop exactly. We can break up his path by the time he spends on the two different parts of his trip, on the sidewalk versus on the grass:
$$t = t_s + t_g$$Where $t_s$ and $t_g$ are the time on the sidewalk and grass, respectively. Recall that we can relate time, distance, and speed by the simple expression $x = vt$, distance is velocity times time. Plugging this into our expression above, we get:
$$t = \frac{x_s}{v_s} + \frac{x_g}{v_g}$$ We are given the speeds that John can walk across the sidewalk and grass and our goal is to find $x_s$, so what remains is to see if we can express $x_g$ in terms of $x_s$.

Note that we are told from his starting position that the bus stop is 2000 ft west and 600 ft north. This forms a right triangle allowing us to use the Pythagorean theorem to find the straight-line distance of walking across the grass. Assuming that John travels $x_s$ ft west, then our right triangle has side lengths $2000 - x_s$ and $600$. Via the Pythagorean theorem we get:

$$x_g = \sqrt{(2000 - x_s)^2 + 600^2}$$ We plug this into our expression for time to get:
$$t = \frac{x_s}{v_s} + \frac{\sqrt{(2000 - x_s)^2 + 600^2}}{v_g}$$ The problem gives us that $v_s = 6$, $v_g = 4$ and $t$ in seconds is $t=450$. Plugging everything in, we get:
$$450 = \frac{x_s}{6} + \frac{\sqrt{(2000 - x_s)^2 + 600^2}}{4}$$

With a bit of algebra, this simplifies to solving the straightforward quadradic:
$$x_s^2 - 2880x_s + 2016000 = 0$$
Which yields two solutions:
$$x_s = 1200\qquad x_s = 1680$$

Both of these solutions are admissable as they are realistic. The intuition behind there being two solutions is that he has two strategies here -- minimize total overall distance by cutting across the grass earlier or spend more time walking faster along the sidewalk, but at a greater cost of distance. In either case, you can find a solution that takes 7 min 30 secs.

For the sake of completeness, we will take $x_s = 1200$. We can double check that this solution works by noting that $x_g = \sqrt{800^2 + 600^2} = 1000$ and $t = 1200/6 + 1000/4 = 200 + 250 = 450$, as desired.