The shape of a tire is determined by its internal pressure and the weight it is bearing. As the weight is increased, the tire flattens against the road surface. This decreases its volume a little and increases its internal pressure, but primarily, this increases the contact area between the tire and the road.
The tire's job is to transmit the force of the weight it is bearing to the road. The weight force can be measured by multiplying the pressure by the contact area. Thus, we can estimate the weight a tire is bearing, by multiplying the tire pressure by the contact area. If my bicycle tire is set to a pressure of 30 pounds per square inch, and I weigh 180 pounds, then I can expect that the two tires together must cover an area of at least 6 square inches, just holding me up.
Detective Kim worked the same way. The discarded tire gave the tire pressure. Since the blood stained an area of the road that included several tires, the contact area of a single tire could be determined. Since the truck was a 16 wheeler, but one wheel was off, an estimate for the weight of the truck would be
weight = 15 * tire pressure * contact area of one tire.
This doesn't include
the weight of the cab, and doesn't account for the small increase
in pressure as the tire is deformed. But this estimate by itself
was over the limit that Lieutenant Lee had reported, so Detective
Kim knew the truck must be stopped.
If you're concerned about the fact that the pressure is not a constant, you might find it interesting to try to work out the formula for the pressure as a function of the load. To keep things simple, assume a single tire, which is has a circular profile under zero load. Assume that as the load increases, the shape of the tire can be modeled by simply slicing off a bit of the bottom. In other words, the part of the tire that is not in touch with the road does not change shape at all. (This can't be true for large loads!) Since the volume decreases, the pressure must increase. Can you determine the formula?
To compute the volume of the tire, it's enough to know how to compute the volume of a circle with a part "sliced away". And to do that, it's enough to know how to compute the volume of the piece sliced away. ...And to do that, you should look at the Watertank puzzle.
Back to The Hijacked Truck Puzzle.