Answer to October 8, 2001 Problem

The BUDGET Truck Problem
A Budget Rental Truck is 12 feet 8 inches tall (from the road to the top of the Van). The cab of the truck is 8 feet long and the van is 24 feet long (see diagram below). If the front of the truck is 8 feet from a stoplight suspended 18 feet above the road, how far behind the truck must the driver of a car be so that he can see the stoplight? Assume that the eye of the driver is 4 feet above the road.

Solution:
The answer is 65 feet!

This problem could be solved with geometry using similar triangles or it could be solved with trigonometry.

In the diagram below, L represents the light (which is 18 feet above the road), T represents the back of the truck (which is 8 + 8 + 24 = 40 feet from the light and 12 feet 8 inches above the road), and E represents the eye of the driver in the car (which is 4 feet above the road and x feet behind the truck). You are solving for x.

The three triangles in the diagram are similar, so you can set up the following proportion:

Using the first two parts of this proportion, you obtain:
4x + 4y = 12.67y

Using the first and last parts of this proportion, you obtain:
18y = 160 + 4x + 4y

Solving simultaneously, you obtain y = 30 feet and x = 65 feet.

Using trig, you could solve for the measure of the acute angle which is common to all three triangles.
The angle = Arctan (5.33 / 40) = 7.59 degrees.
Then substituting in the equation
tan (7.59) = 8.67 / x,
you again obtain 65 feet for the value of x.


Keith Mealy wrote:
I think you forgot to add a second part to this problem:

If driver A actually stays that far behind the rental truck at a light, how long before driver B (the vehicle immediately behind driver A) honks his horn to move up?

a) (Boston) 0.000007 seconds
b) (New York) will not beep, but stick head out window and yells (HEY, *#&$HEAD, MOVE THE *#&@ UP!!!! WHADDYA DOING, YOU #&*()@*)!!!! )
c) (Washington DC) Will just pass on the shoulder.
d) (Los Angeles) Will just pass on one of the other 5 lanes in the same direction.
e) (Chicago) What light?
f) (Philadelphia) Will cross double yellow and cut in in front of driver A, while giving appropriate gesticulation.


Correctly solved by:

1. Keith Mealy Cincinnati, Ohio
2. Renata Sommerville Austin, Texas
3. Richard K. Johnson La Jolla, California
4. Kirstine Wynn Winchester, Virginia
5. Walt Arrison Philadelphia, Pennsylvania
6. Diana Xing Fort Collins, Colorado
7. Nick von Keller Winchester, Virginia
8. Joe Heintz Manchester, Tennessee
9. Tony Wu Fort Collins, Colorado
10. Janine Oliver Winchester, Virginia
11. Jerry Putt Loudoun Valley, Virginia
12. Geoff Keith Santa Monica, California
13. Brendan Jones Winchester, Virginia