Problem 74 - Prealgebra

Question

Vehicle Weight If you can measure the area that the tires on your car contact the ground, and you know the air pressure in the tires, then you can estimate the weight of your car, in pounds, with the following formula: $$W=A P N$$ where \(W\) is the vehicle's weight in pounds, \(A\) is the average tire contact area with a hard surface in square inches, \(P\) is the air pressure in the tires in pounds per square inch (psi, or \(\mathrm{lb} / \mathrm{in}^{2}\) ), and \(N\) is the number of tires.(IMAGE CANNOT COPY) a. What is the approximate weight of a car if the average tire contact area is a rectangle 6 inches by 5 inches and if the air pressure in the tires is 30 psi?

Step-by-Step Solution

Answer

The approximate weight of the car is 3600 pounds.

Step 1: Calculate the Tire Contact Area

First, we need to find the average tire contact area in square inches. It is given that the contact area is a rectangle measuring 6 inches by 5 inches. The area, \( A \), can be calculated using the formula for the area of a rectangle: \( A = \, \text{length} \times \text{width} \). So, \( A = 6 \times 5 = 30 \; \text{square inches} \).

Step 2: Identify the Number of Tires

Typically, a car has 4 tires, which means \( N = 4 \) tires.

Step 3: Use the Formula to Calculate Vehicle Weight

We use the formula \( W = A P N \) to calculate the vehicle's weight. We know \( A = 30 \; \text{square inches} \), \( P = 30 \; \text{psi} \), and \( N = 4 \). Therefore, the weight \( W \) is: \( W = 30 \times 30 \times 4 = 3600 \; \text{pounds} \).

Key Concepts

Understanding Tire Contact Area

The tire contact area refers to the part of the tire that is in direct touch with the ground. It’s crucial because it influences the distribution of the car's weight across its tires and affects handling and safety. The tire contact area can be visualized as the footprint of the tire on the ground.

The size of this contact area depends on various factors, including the tire’s design, the car’s load, and the air pressure inside the tire. In essence, it’s the actual spot where the rubber meets the road.

To calculate the tire contact area, you can multiply the length and width of the tire’s footprint. In the example, if the footprint of the tire measures 6 inches by 5 inches, the contact area is:

• Length = 6 inches
• Width = 5 inches

The area, denoted as \( A \), is calculated as:\[ A = 6 \times 5 = 30 \; \text{square inches} \]
This measurement is essential in various calculations, especially when determining the overall pressure exerted by a vehicle.

Exploring Air Pressure in Tires

Air pressure in tires is vital because it affects both safety and efficiency. Measured in psi (pounds per square inch), proper air pressure ensures that the tire maintains its preferred shape and performs well across various conditions.

When the tire pressure is too low, more of the tire touches the ground, which can increase rolling resistance and cause the tire to wear out faster. On the other hand, high pressure results in less contact, potentially reducing traction and affecting the ride quality.

In our exercise, the air pressure in the tires is given as 30 psi. This means that every square inch of the tire is being pushed outward with a force of 30 pounds. Proper maintenance of tire pressure using a pressure gauge can extend a tire's lifespan and improve fuel efficiency, ensuring that the data used in calculations, like vehicle weight estimation, are accurate and reliable.

Understanding the Vehicle Weight Formula

The vehicle weight formula is a straightforward yet effective way to estimate a car's weight using the tire contact area, tire pressure, and the number of tires.

The formula is given by:\[ W = A \times P \times N \]where:

• \( W \) = Vehicle weight in pounds
• \( A \) = Average tire contact area in square inches
• \( P \) = Air pressure in psi
• \( N \) = Number of tires

This formula helps us understand how the car's weight is supported and distributed by its tires. By knowing the tire’s contact area and the air pressure, alongside the number of tires, this method provides a rough estimation of a vehicle's weight.

In our example:

• \( A = 30 \) square inches
• \( P = 30 \) psi
• \( N = 4 \) tires

Thus, the weight \( W \) is calculated as:\[ W = 30 \times 30 \times 4 = 3600 \; \text{pounds} \]This result shows how these factors combine to support the weight of a vehicle, using a simple multiplication of contact area, pressure, and tire count.