To solve this problem, we need to calculate the total number of vehicles (cars and trucks) stuck in the traffic jam. Here's the step-by-step solution:

Step 1: Convert the Length of the Traffic Jam to Feet

The total length of the traffic jam is given as 8 miles. Since there are 5280 feet in a mile:

$\text{Length of traffic jam in feet} = 8 \times 5280 = 42,240 \text{ feet}$

Step 2: Calculate the Average Length of a Car and a Truck with the Distance Between Vehicles

Let's compute the total average space taken by each car and each truck, including the 3-foot gap between vehicles:

• The average length of a car including the gap is: $\text{Length of a car including gap} = 13.5 + 3 = 16.5 \text{ feet}$

• The average length of a truck including the gap is: $\text{Length of a truck including gap} = 20 + 3 = 23 \text{ feet}$

Step 3: Use the Percentages to Express the Total Length Covered by Cars and Trucks

Since 75% of the vehicles are cars and 25% are trucks, let’s denote:

• $x$ = number of cars
• $y$ = number of trucks

We can express the total length of the traffic jam in terms of the cars and trucks:

$16.5x + 23y = 42,240$

We also know that 75% of the vehicles are cars and 25% are trucks, so:

$x = 0.75(x + y), \quad y = 0.25(x + y)$

Step 4: Solve for the Number of Vehicles

From the equation $x = 0.75(x + y)$, we can solve for $y$:

$y = \frac{x}{3}$

Substitute this into the total length equation:

$16.5x + 23 \left(\frac{x}{3}\right) = 42,240$

Now, simplify and solve for $x$:

$16.5x + \frac{23x}{3} = 42,240$

Multiply everything by 3 to eliminate the fraction:

$3 \times 16.5x + 23x = 126,720$

$49.5x + 23x = 126,720$

$72.5x = 126,720$

$x = \frac{126,720}{72.5} \approx 1,748$

Now, find $y$:

$y = \frac{x}{3} = \frac{1,748}{3} \approx 582.67$

Since the number of trucks must be a whole number, round to the nearest whole number:

$y \approx 583$

Step 5: Calculate the Total Number of Vehicles

Total number of vehicles $N$ is:

$N = x + y = 1,748 + 583 = 2,331$

Conclusion

There are approximately 2,331 vehicles stuck in the traffic jam.

Would you like more details or have any questions about this solution?

Related Questions:

1. What would happen to the total number of vehicles if the distance between vehicles increased to 5 feet?
2. How would the calculation change if the percentage of cars and trucks were reversed?
3. What is the total length in feet occupied only by the trucks?
4. How many trucks would there be if the average truck length increased by 10%?
5. What if the traffic jam was 12 miles long instead of 8 miles?

Tip: Remember to always include the average gap distance when calculating total lengths in traffic problems!