Solve each problem. See Examples 5 and 6. Formaldehyde is an indoor air pollutant formerly found in plywood, foam insulation, and carpeting. When concentrations in the air reach 33 micrograms per cubic foot (μg/ft^3), eye irritation can occur. One square foot of new plywood could emit 140 μg per hr. (Data from A. Hines, Indoor Air Quality & Control.) The room contains 800 ft^3 of air and has no ventilation. Determine how long it would take for concentrations to reach 33 μg/ft^3. (Round to the nearest tenth.)

Verified step by step guidance

1. Identify the known quantities: the emission rate of formaldehyde is 140 micrograms per hour (μg/hr), the volume of the room is 800 cubic feet (ft^3), and the target concentration is 33 micrograms per cubic foot (μg/ft^3).
2. Set up the relationship between the total amount of formaldehyde emitted, the volume of the room, and the concentration. The concentration (C) is given by the total amount of formaldehyde (A) divided by the volume (V): \(C = \frac{A}{V}\).
3. Express the total amount of formaldehyde emitted as a function of time (t) in hours. Since the emission rate is 140 μg/hr, the total amount emitted after time t is \(A = 140t\).
4. Substitute \(A = 140t\) and \(V = 800\) into the concentration formula to get \(33 = \frac{140t}{800}\), which relates time t to the concentration.
5. Solve the equation for t by multiplying both sides by 800 and then dividing by 140 to isolate t: \(t = \frac{33 \times 800}{140}\). This will give the time in hours needed to reach the concentration of 33 μg/ft^3.

Key Concepts

Concentration Calculation

Concentration is the amount of a substance per unit volume. In this problem, concentration (μg/ft³) is found by dividing the total amount of formaldehyde emitted (μg) by the volume of air (ft³). Understanding this relationship helps determine when the pollutant reaches a harmful level.

Rate of Emission and Time Relationship

The rate of emission (μg/hr) indicates how much formaldehyde is released per hour. By knowing the emission rate and the target concentration, you can calculate the time needed for the pollutant to accumulate to that concentration in a closed space without ventilation.

Unit Conversion and Dimensional Analysis

Properly handling units like micrograms, cubic feet, and hours is essential. Dimensional analysis ensures that calculations are consistent and meaningful, allowing you to convert between total emitted mass, concentration, and time accurately.