Solve each problem. (Modeling) Lead Intake As directed by the 'Safe Drinking Water Act' of December 1974, the EPA proposed a maximum lead level in public drinking water of 0.05 mg per liter. This standard assumed an individual consumption of two liters of water per day. If EPA guidelines are followed, write an equation that models the maximum amount of lead A ingested in x years. Assume that there are 365.25 days in a year.

Verified step by step guidance

1. Identify the given information: the maximum lead concentration is 0.05 mg per liter, and the individual consumes 2 liters of water per day.
2. Calculate the maximum amount of lead ingested per day by multiplying the concentration by the daily water intake: \(0.05 \times 2\) mg per day.
3. Determine the number of days in x years using the given value of 365.25 days per year: \(365.25 \times x\) days.
4. Write an expression for the total amount of lead ingested in x years by multiplying the daily lead intake by the total number of days: \((0.05 \times 2) \times (365.25 \times x)\) mg.
5. Simplify the expression to form the equation modeling the maximum amount of lead ingested in x years: \(A = 0.05 \times 2 \times 365.25 \times x\).

Key Concepts

Unit Conversion and Dimensional Analysis

Unit conversion involves changing quantities from one unit to another to maintain consistency in calculations. In this problem, converting daily lead intake (mg per liter and liters per day) into total lead intake over years requires careful multiplication of units like mg, liters, days, and years to ensure the final amount is correctly expressed.

Linear Modeling

Linear modeling uses a linear equation to represent a relationship between variables. Here, the total lead intake A is modeled as a linear function of time x (in years), assuming constant daily intake. The equation will multiply daily intake by the number of days in x years, showing direct proportionality.

Use of Constants and Assumptions in Modeling

Mathematical modeling often relies on constants and assumptions to simplify real-world scenarios. The problem assumes a fixed daily water consumption (2 liters) and a fixed number of days per year (365.25) to account for leap years, which are essential for accurately calculating total lead intake over multiple years.