Solve each problem. Force of WindThe force of the wind blowing on a vertical surface varies jointly as the area of the surface and the square of the velocity. If a wind of 40 mph exerts a force of 50 lb on a surface of 1/2 ft^2, how much force will a wind of 80 mph place on a surface of 2 ft^2?

Key Concepts

Joint Variation

Joint variation occurs when a quantity varies directly with the product of two or more other quantities. In this context, the force of the wind is directly proportional to both the area of the surface and the square of the wind's velocity. This relationship can be expressed mathematically as F = k * A * V^2, where F is the force, A is the area, V is the velocity, and k is a constant of proportionality.

Proportionality Constant

The proportionality constant, often denoted as 'k', is a value that relates the variables in a joint variation equation. It is determined by substituting known values into the equation. In this problem, the constant can be calculated using the initial conditions provided (40 mph wind, 50 lb force, and 1/2 ft² area) to find 'k', which will then be used to solve for the unknown force under different conditions.

Quadratic Relationships

Quadratic relationships involve variables that are squared, leading to a parabolic relationship between them. In this scenario, the wind's velocity is squared, meaning that if the velocity doubles, the force exerted increases by a factor of four, assuming the area remains constant. Understanding this concept is crucial for accurately calculating the force exerted by the wind at different velocities.