Question

The consumption of Swiss cheese in the United States from 1970 to 1996 can be modeled by \(P=-0.002 t^{2}+0.056 t+0.889\) where \(P\) is the number of pounds per person and \(t\) is the number of years since \(1970 .\) According to the graph of the model, in what year would the consumption of Swiss cheese drop to \(0 ?\) Is this a realistic prediction? (GRAPH CANNOT COPY).

Short answer

The exact answer can be computed using the quadratic formula. Evaluate the feasibility by comparing the resulting year to history and industry's trends in cheese consumption.

Step-by-step solution

Identify Important Information

The equation \(P=-0.002 t^{2}+0.056 t+0.889\) illustrates the consumption of Swiss Cheese in the United States. Here, \(P\) corresponds to the quantity of cheese consumed per person (in pounds), and \(t\) signifies the number of years since 1970.

Set up and Solve the Equation

To determine the year when cheese consumption becomes zero, set \(P\) to \(0\) in the equation and solve for \(t\). This can be done by equalizing the polynomial to zero and finding its roots: \(0 = -0.002 t^{2}+0.056 t+0.889\).

Calculate Roots using Quadratic Formula

The quadratic equation can have zero, one, or two solutions, depending on the discriminant. The quadratic formula, \(t = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\), can be used to calculate the roots, where \(a=-0.002\), \(b=0.056\), and \(c=0.889\).

Analyze the Result

The roots obtained will represent the years (since 1970) when the cheese consumption drops to zero. However, each solution must be checked to ensure they fall within the initial model’s limits.

Reality Check

Determining whether the years derived are feasible entails a quick reality check. Analyzing the realistic prediction is historical and based on the growth and changes seen in the cheese industry.