Garfield and Odie are situated at (0,0) and (25,0), respectively. Suddenly, Garfield and Odie dash in the direction of the point (9, 12) at speeds of 7 and 10 units per minute, respectively. During this chase, find the minimum distance between Garfield and Odie.
Add one more point, (9,0). Then drawing the altitude from (9,12) to (9,0). This creates a right triangle for each of Garf and Odie.
Lets look at Garf. His triangle is has sides 9, 12, and 15. If he travels 1 unit along the hypotenuse then his x-axis displacement is 3/5 unit and his y-axis displacement is 4/5 unit.
So then with a speed of 7 units then I can parameterize Garf's location as ((21/5)*t, (28/5)*t).
Now for Odie. His triangle is has sides 16, 12, and 20. If he travels 1 unit along the hypotenuse then his x-axis displacement is -4/5 unit and his y-axis displacement is 3/5 unit.
So then with a speed of 10 units then I can parameterize Odie's location as (25-8*t, 6*t).
Then the distance D between them can be expressed using the distance formula: D^2 = (25-(61/5)*t)^2 + ((2/5)*t)^2 = 149t^2 - 610t + 625
The final expression for D^2 is a parabola, so it is quick to find its vertex is at 610/(149*2) = 305/149 ~= 2.0470.
Then completing the square gives D^2 = 149*(t-305/149)^2+625-305^2/149. So at t=305/149 then D^2 = 625-305^2/149 ~= 0.6711, then this makes D ~= 0.8192 units.
Analytic Solution
