Problem 95 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

U.S. NURSING SHORTAGE The demand for nurses between 2000 and 2015 is estimated to be $$ D(t)=0.0007 t^{2}+0.0265 t+2 \quad(0 \leq t \leq 15) $$ where \(D(t)\) is measured in millions and \(t=0\) corresponds to the year 2000 . The supply of nurses over the same time period is estimated to be $$ S(t)=-0.0014 t^{2}+0.0326 t+1.9 \quad(0 \leq t \leq 15) $$ where \(S(t)\) is also measured in millions. a. Find an expression \(G(t)\) giving the gap between the demand and supply of nurses over the period in question. b. Find the interval where \(G\) is decreasing and where it is increasing. Interpret your result. c. Find the relative extrema of \(G\). Interpret your result.

Step-by-Step Solution

Answer

The gap function between the demand and supply of nurses is given by \(G(t) = 0.0021t^2 - 0.0061t + 0.1\). The function is decreasing in the interval \(0 \leq t < 1.45\), meaning the gap narrows from 2000 to around 2001/2002. The function is increasing in the interval \(1.45 \leq t \leq 15\), meaning the gap widens from 2001/2002 until 2015. The smallest gap (local minimum) occurs at the point \((1.45, 0.08431)\), which is around 2001/2002.

Step 1: Calculate the Gap Function: G(t)

\(G(t) = D(t) - S(t)\) Plug the given expressions for \(D(t)\) and \(S(t)\) into the above equation and simplify: \(G(t) = (0.0007t^2 + 0.0265t + 2) - (-0.0014t^2 + 0.0326t + 1.9)\) \(G(t) = 0.0007t^2 + 0.0265t + 2 + 0.0014t^2 - 0.0326t - 1.9\) Add and combine like terms: \(G(t) = (0.0007t^2 + 0.0014t^2) + (0.0265t - 0.0326t) + (2 - 1.9)\) \(G(t) = 0.0021t^2 - 0.0061t + 0.1\) #b. Increasing and Decreasing Intervals of G(t)# To find the intervals where G(t) is increasing or decreasing, we need to determine the critical points by finding the first derivative of G(t) and setting it equal to 0.

Step 2: Calculate the First Derivative of G(t)

\(G'(t) = \frac{d}{dt} (0.0021t^2 - 0.0061t + 0.1)\) To find the first derivative, differentiate G(t) term by term: \(G'(t) = \frac{d}{dt} (0.0021t^2) - \frac{d}{dt} (0.0061t) + \frac{d}{dt}(0.1)\) \(G'(t) = 0.0042t - 0.0061\) Now we will find the critical points by setting the first derivative G'(t) equal to 0 and solve for t.

Step 3: Find Critical Points

\(0.0042t - 0.0061 = 0\) To solve for t, first, isolate t: \(t = \frac{0.0061}{0.0042}\) \(t \approx 1.45\) Now we need to determine if this critical point corresponds to an increasing or decreasing interval by applying the first derivative test. We will use this critical point to divide the domain into two intervals and test a value from each interval to see if the function is increasing or decreasing in that interval.

Step 4: First Derivative Test

For interval \(0 \leq t < 1.45\), let's choose test point \(t = 1\): \(G'(1) = 0.0042(1) - 0.0061 = -0.0019 < 0\) Since the first derivative at \(t = 1\) is negative, G(t) is decreasing in the interval \( 0 \leq t < 1.45\). For interval (1.45) ≤ t < 15], let's choose test point \(t = 2\): \(G'(2) = 0.0042(2) - 0.0061 = 0.0023 > 0\) Since the first derivative at \(t = 2\) is positive, G(t) is increasing in the interval \(1.45 \leq t \leq 15\). So G(t) is decreasing in the interval \(0 \leq t < 1.45\) and increasing in the interval \(1.45 \leq t \leq 15\). This result means that the gap between the demand and supply of nurses narrows from 2000 to around 2001/2002 before widening until 2015. #c. Relative Extrema of G(t)# The critical point found in part b is the point where the function changes from decreasing to increasing. It corresponds to a local minimum. We can find the local minimum of the function by plugging in the critical point, t = 1.45, into the function G(t).

Step 5: Find Local Minimum

\(G(1.45) = 0.0021(1.45)^2 - 0.0061(1.45) + 0.1\) Calculate: \(G(1.45) \approx 0.08431\) So, there is a local minimum of G(t) at (1.45, 0.08431), which means that the smallest gap between the demand and the supply of nurses during the given time period occurs around 2001/2002.

Key Concepts

Gap Function in Applied Mathematics

The concept of a gap function is central to studies involving two competing sets of data or requirements, such as supply and demand in economics. A gap function identifies the difference between these two sets and it is expressed as a straightforward subtraction of one function from the other. For example, in nursing dynamics, the gap function, denoted as \( G(t) = D(t) - S(t) \), measures the difference between the demand for nurses, denoted as \( D(t) \), and the supply of nurses, \( S(t) \). Understanding the behavior of this gap over time helps in planning and policy-making within the healthcare sector.

In the given exercise, we calculate the gap function to understand how the demand and supply of nurses change over time. This simple mathematical model can help policymakers identify periods of shortage or surplus of nursing resources.

Critical Points in Calculus

In calculus, critical points are where a function's derivative is either zero or undefined. At these points, the function's graph can experience a shift in direction, and this is where you might find local maxima or minima (high or low points). To find a function’s critical points, we take the derivative of the function and set it to zero.

For instance, the first derivative of the gap function \( G'(t) \) in our nursing example is set to zero to find the values of \( t \) at which the rate of change of the gap is stationary. Critical points are vital in analyzing the function's behavior because they help us determine where the graph changes course, which in the context of nursing, could mean identifying when nursing shortage begins to improve or worsen.

Increasing and Decreasing Intervals

Increasing and decreasing intervals on a function’s graph represent where the function is rising or falling, respectively. Specifically, a function is increasing on an interval if, for any two points in that interval, a larger input leads to a larger output. Conversely, it is decreasing if a larger input leads to a smaller output.

Returning to our nursing gap example, we use the first derivative test to examine intervals around critical points and define where the gap function is increasing or decreasing. This allows us to understand the periods when the shortage of nurses is getting worse (increasing intervals) or improving (decreasing intervals), which holds significant implications for healthcare management.

Relative Extrema

Relative extrema refer to the local highs and lows of a function, which are found at certain critical points. A relative maximum is where a function's value is higher than all nearby points, while a relative minimum is where it’s lower than all neighboring points. Calculating the function’s value at these points provides the precise extremum.

In the scenario given, we find that the local minimum of our gap function corresponds to the point in time where the gap between the demand and the supply of nurses was smallest. Identifying this minimum is crucial for health service administration as it marks a period where the staffing situation was most balanced and can help to analyze and plan the required nurse staffing levels.