An old problem from Popular Mechanics

A car always travels at the same speed s miles per minute from point A to point C. There are 3 routes:_

(1.) straight to C. This takes 30 minutes,

(2.) go north 10 miles to B, then head straight to C. This takes 35 minutes, or

(3.) go east 10 miles to D, then head straight to C for 40 minutes.

If angle BAD is 90 degrees, what is the speed of the car?

One Solution Set up a coordinate system with $A=(0,0)$, $B=(0,10)$, $C=(x,y)$, $D=(-10,0)$. Using distance = rate x time, we get three equations in 3 unknowns $s$, $x$, $y$:

(1) AC = $\sqrt{x^2 + y^2}=30 s$, (2) ABC = $10+\sqrt{x^2+(y-10)^2}=35 s$, (3) ADC = $10+\sqrt{(x+10)^2+y^2}=40 s $.

Solving first for $s$, we get two possible times: $s_1=-\frac{8}{953} \, \sqrt{6461} + \frac{1260}{953} \approx 0.64738$ miles per minute or $38.843$ miles per hour and $s_2=\frac{8}{953} \, \sqrt{6461} + \frac{1260}{953} \approx 1.9969$ miles per minute or $119.81$ miles per hour.

This gives two possible locations for the point C: $C_1 \approx \left(-11.23,\,15.85\right)$ when $s=s_1$ and $C_2 \approx \left(59.69,\,5.093\right)$ when $s=s_2$.

The quadrilateral $ABC_1D$ is nice and convex. The quadrilateral $ABC_2D$ is not, but it still satisfies the statement of the problem (no speed limit was imposed). So there are two solutions to this problem.

Below is the code in Sagemath used to calculate the solutions.

This problem, like any problem, can be parameterized making any one or more of the specific numbers in the problem a parameter and asking the same or related question. Thus this one problem can give rise to a host of similar problems.

For example, suppose that we make the time straight from A to C a parameter $ t_1$. Now the three equations in $ x,y$ and $s$ have a parameter $t_1$ in place of a number 30. We can still solve for $x$, and $y$ in terms of $s$, but $s$ will have a parameter $t_1$. Then another problem presents: For what values of $t_1$ does the problem have a solution? We know when $t_1=30$, there are two solutions. We can modify our computations above.

Looking at these graphs, we see that $s_1 \lt s_2$ are reasonable speeds for any time between 3.6 and 35 minutes. Now let's look at the graphs for the destinations $C_1 = (x_1,y_1)$ and $C_2 = (x_2,y_2)$ for the speeds $s_1$ and $s_2$.

Here is a sagelet to play with various values for $t_1$ (the time the car takes to go straight to C). Change the value for $t_1$ and press Update.

Try adding another parameter to the problem: For example let the distance from A to B be b, instead of 10.

Use this sagelet to explore the speed s of the car as a function of b (the vertical distance from A to B) and t1 (the time it takes the car to go straight from A to C)