Problem 106 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Clark County in Nevada-dominated by greater Las Vegas-is one of the fastest- growing metropolitan areas in the United States. The population of the county from 1970 through 2000 is approximated by the function $$ \begin{array}{r} P(t)=44560 t^{3}-89394 t^{2}+234633 t+273288 \\ (0 \leq t \leq 4) \end{array} $$ where \(t\) is measured in decades, with \(t=0\) corresponding to the beginning of 1970 . a. Show that the population of Clark County was always increasing over the time period in question. Hint: Show that \(P^{\prime}(t)>0\) for all \(t\) in the interval \((0,4)\). b. Show that the population of Clark County was increasing at the slowest pace some time toward the middle of August 1976 . Hint: Find the inflection point of \(P\) in the interval \((0,4)\).

Step-by-Step Solution

Answer

The population of Clark County was always increasing from 1970 through 2000 as \(P'(t) > 0\) for all \(t\) in the interval \((0, 4)\). Additionally, the population growth was at its slowest pace sometime towards the middle of August 1976, as indicated by the inflection point of the population function.

Step 1: Find the first derivative

Differentiate the given function with respect to \(t\): \[ P'(t) = \frac{dP}{dt} = 3(44560)t^2 - 2(89394)t + 234633 \]

Step 2: Show that the first derivative is always positive

Now, we want to show that \(P'(t) > 0\) for all \(t\) in the interval \((0, 4)\). To find where the first derivative is positive, we'll first find its critical points: \[P'(t) = 0\] \[3(44560)t^2 - 2(89394)t + 234633 = 0\] Let's simplify the expression: \[ 133680t^2 - 178788t + 234633 = 0 \] The above is a quadratic equation. To check if it has any real roots in the interval \((0, 4)\), we can calculate the discriminant. \[D = b^2 - 4ac\] \[D = (-178788)^2 - 4(133680)(234633)\] \[D = 31952677424 - 125726694400\] \[D = -93774016976\] We notice that the discriminant, D, is negative, which implies that the quadratic equation has no real roots. Hence, we can conclude that \(P'(t) > 0\) for all \(t\) in the interval \((0, 4)\), and the population function is always increasing over the time period in question.

Step 3: Find the second derivative

Differentiate the first derivative with respect to \(t\): \[ P''(t) = \frac{d^2P}{dt^2} = 6(44560)t - 2(89394) = 267360t - 178788 \]

Step 4: Find the inflection point

To find the inflection point, equate the second derivative to zero and solve for \(t\): \[P''(t) = 0\] \[267360t - 178788 = 0\] \[t = \frac{178788}{267360}\] \[t \approx 0.6685\] The above value of \(t\) is in the interval \((0, 4)\). Now we need to find the corresponding time closest to the middle of August 1976. Since \(t\) is in decades, we can convert this value to years by multiplying by 10: \[0.6685\times 10 = 6.685 \text{ years} \] Now, since \(t = 0\) corresponds to the beginning of 1970, we can add this time to find out when the slowest growth in population occurred: \[1970 + 6.685 \approx 1976.685\] The time, rounded to the nearest month, is August 1976. Hence, the population of Clark County was increasing at the slowest pace sometime towards the middle of August 1976.

Key Concepts

Differential Calculus

Differential calculus is a fundamental tool in mathematical analysis that deals with the calculation of the rates at which quantities change. It is the process of finding the derivative of a function, which represents the change in the dependent variable with respect to an independent variable.In the Clark County population growth problem, differential calculus allows us to understand the rate at which the population changes over time. By taking the first derivative of the population function, denoted as \( P'(t) \), we can determine if the population is increasing or decreasing at any point during the specified interval.An important aspect is to determine the sign of the first derivative. If \( P'(t) > 0 \) throughout an interval, it suggests that the population is consistently increasing over that interval. By applying this analysis to the population function of Clark County, the solution clearly demonstrates that the population was always growing from 1970 to 2000 since the first derivative was always positive.

Inflection Points

Inflection points are significant in the study of a function's graph because they represent points where the curve changes its concavity. In other words, an inflection point is where the function switches from being concave upward to concave downward, or vice versa.To find the inflection points of a function, you need to determine where the second derivative of the function, represented by \( P''(t) \), changes sign. The solution to the exercise shows that after calculating the second derivative of the population function, setting \( P''(t) \) equal to zero gives the point in time where the rate of population growth shifts.The presence of an inflection point tells us not only that the concavity has changed but also often identifies intervals of acceleration and deceleration in growth. For Clark County, finding the inflection point helped establish when the population was increasing at the slowest pace, which is a crucial piece of information for urban planning and resource allocation.

Quadratic Equations

Quadratic equations are a form of polynomial equations that follow the standard form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a \) is not equal to zero. These equations are crucial in various areas of algebra and are especially known for their characteristic 'U'-shaped graphs called parabolas.In analyzing population growth through differential calculus, when we derive the first derivative of the population function and equate it to zero to find critical points, it often results in a quadratic equation. The discriminant, \( D = b^2 - 4ac \), then plays a pivotal role in determining the nature of the roots of the equation.A negative discriminant, such as what we encountered in the Clark County population growth analysis, signifies that there are no real solutions to the equation. This implies that the first derivative does not cross the x-axis and hence, the population function does not have a maximum or minimum within the interval—confirming that the population was increasing throughout the given time frame.