Problem 38 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

ENVIRONMENTAL ANALYSIS An environmental study of a certain community suggests that the average daily level of smog in the air will be \(Q(p)=\sqrt{0.5 p+19.4}\) units when the population is \(p\) thousand. It is estimated that \(t\) years from now, the population will be \(p(t)=8+0.2 t^{2}\) thousand. a. Express the level of smog in the air as a function of time. b. What will the smog level be 3 years from now? c. When will the smog level reach 5 units?

Step-by-Step Solution

Answer

a. \(Q(t) = \sqrt{23.4 + 0.1t^2}\). b. Approximately 4.93 units. c. In 4 years.

Step 1: Express the Population as a Function of Time

First, recall the population function: \[ p(t) = 8 + 0.2t^2 \]This function describes the community's population in thousands as a function of time (years).

Step 2: Substitute the Population Function into the Smog Function

Next, substitute the population function into the smog function:\[ Q(p) = \sqrt{0.5p + 19.4} \]Since \( p = p(t) = 8 + 0.2t^2 \), we get:\[ Q(t) = \sqrt{0.5(8 + 0.2t^2) + 19.4} \]

Step 3: Simplify the Equation

Simplify the expression obtained to make the function of time explicit:\[ Q(t) = \sqrt{0.5(8) + 0.5(0.2t^2) + 19.4} = \sqrt{4 + 0.1t^2 + 19.4} = \sqrt{23.4 + 0.1t^2} \]Thus, the smog level as a function of time is:\[ Q(t) = \sqrt{23.4 + 0.1t^2} \]

Step 4: Find the Smog Level 3 Years from Now

Substitute \( t = 3 \) into the smog level function:\[ Q(3) = \sqrt{23.4 + 0.1(3)^2} = \sqrt{23.4 + 0.9} = \sqrt{24.3} \approx 4.93 \, \text{units} \]

Step 5: Determine When the Smog Level Will Reach 5 Units

Set \( Q(t) = 5 \) and solve for \( t \):\[ 5 = \sqrt{23.4 + 0.1t^2} \]Square both sides:\[ 25 = 23.4 + 0.1t^2 \]Subtract 23.4 from both sides:\[ 1.6 = 0.1t^2 \]Divide by 0.1:\[ 16 = t^2 \]Take the square root of both sides:\[ t = \sqrt{16} \]\[ t = 4 \]The smog level will reach 5 units in 4 years.

Key Concepts

population function

In the given environmental calculus problem, the first concept we need to understand is the population function. It is represented as:

• \[ p(t) = 8 + 0.2t^2 \]

This function tells us the population size in thousands over time, where \( t \) represents years. The population starts at 8,000 (when \( t = 0 \)). As time (\( t \)) increases, the population grows. The growth is not linear but quadratic because the \( 0.2t^2 \) term has \( t \) squared. This means population growth accelerates over time. Understanding the population function is key to solving the overall problem as it tells us how many people there will be at any given time in the future.

smog level calculation

Next, we need to calculate the smog level based on population. The smog function is given as:

• \[ Q(p) = \sqrt{0.5p + 19.4} \]

The smog level \( Q \) depends on the population \( p \) in thousands. The function explains how smog increases with more people. There are two parts to this formula: a term dependent on the population, \( 0.5p \), and a constant, 19.4. Adding more people increases the pollution component of the smog level calculation. The constant output of 19.4 adds to the baseline smog level. Thus, this function directly connects population size to the amount of smog in the air.

time-dependent functions

Now, let's talk about how these functions change over time. Initially, we have two separate functions:

• The population function, \( p(t) \), that changes with time.
• The smog function, \( Q(p) \), that changes with population.

To connect them, we substitute \( p(t) \) into the smog function so we can describe smog levels directly as a function of time (\( t \)). This step is crucial for solving our questions relating to future smog levels. After substitution, make sure to simplify the resulting function to make it easy to handle. The combined function becomes:

• \[ Q(t) = \sqrt{23.4 + 0.1t^2} \]

This new function helps determine smog levels at any point in time.

applying square roots

In this problem, square roots play an important role, especially in the smog level calculation. The smog function, \( Q(p) \), involves computing the square root:

• \[ Q(p) = \sqrt{0.5p + 19.4} \]

When we substituted \( p(t) \) into this function, we got:

• \[ Q(t) = \sqrt{23.4 + 0.1t^2} \]

Remember when dealing with square roots:

• Ensure the term inside (the radicand) is non-negative.
• Simplify the radicand as much as possible before applying the square root.

Accurately handling square roots is essential in determining the smog levels at various points in time and solving for \( t \).

algebraic substitution

Algebraic substitution is a powerful technique used here to simplify expressions and solve complex equations. We substituted the population function \( p(t) \) into the smog function \( Q(p) \). This turned two separate functions into one time-based function. Steps involved in substitution:

• Identify the variable you want to replace (in our case, population \( p \)).
• Replace it with the corresponding expression (\( p(t) = 8 + 0.2t^2 \)).
• Simplify the new expression.

This leads us to solve specific questions, like finding future smog levels or specific times when smog reaches a certain level. It's a method that helps bridge the relationships between different variables.