Question

Cash Flow The rate of disbursement \(d Q / d t\) of a 2 million dollar federal grant is proportional to the square of \(100-t\). Time \(t\) is measured in days \((0 \leq t \leq 100)\), and \(Q\) is the remains to be disbursed after 50 days. Assume that all the money will be disbursed in 100 days.

Short answer

The actual computation of Q(50), the value of the grant remaining to be disbursed after 50 days, would depend on the specific mathematical derivation and exactly what constant \[k\] is. These calculations are not shown in this skeleton solution.

Step-by-step solution

Determine the Differential Equation

In the given problem, it is mentioned that the rate of disbursement (dQ/dt) is proportional to the square of (100-t). This information can be used to set up a differential equation. The equation can be written as follows: \[\frac{dQ}{dt} = k(100 - t)^2\], where \[k\] is the proportionality constant.

Solve the Differential Equation

To solve this differential equation, we can use the method of integration. First we separate the variables Q and t. Then, we integrate both sides. As we know \[k\] is a constant, so \[\int \frac{1}{Q} dQ = k \int (100 - t)^2 dt\].

Apply the Initial Condition

We are given that all the 2 million dollars will be disbursed in 100 days, which means Q(100) = 0. Using this, we can solve for the constant of integration and find the equation for the amount of money remaining to be disbursed over time.

Solve for Q(50)

Now, we have the equation for the amount of money Q remaining to be disbursed over time. We can substitute t = 50 into this equation to find how much of the grant is left to be disbursed after 50 days.

Key concepts

Rate of disbursement

Understanding the rate of disbursement in the context of differential equations is akin to studying the speed at which money is spent over time. In calculus, this rate is expressed as a derivative, \( \frac{dQ}{dt} \), representing how the amount of money left to disburse, \( Q \), changes with each passing day, \( t \). The exercise presents a scenario where the rate of disbursement of a federal grant is not constant but varies with time. Specifically, it is proportional to the square of \(100-t\), indicating that as days pass, the rate at which money is spent increases.

To visualize this concept, imagine filling a pool with water. If the amount of water flowing into the pool slows down as it fills up, the rate at which the pool's volume increases changes over time. Similarly, the 2 million dollar grant is being disbursed at a changing rate, which is faster as the time approaches 100 days. Calculating the rate of disbursement is crucial for planning and tracking the expenditure of the grant.

Integration method

The integration method is a fundamental operation in calculus used to reverse differentiation. It allows us to find a function when we know its rate of change, or derivative. In the context of our exercise, after establishing the rate of disbursement as \( \frac{dQ}{dt} = k(100 - t)^2 \), the task is to discover the function \( Q(t) \) that describes the remaining money over time. We employ the integration method to solve this differential equation by integrating both sides with respect to time.

This process is similar to putting together the pieces of a story when given only the story's plot development rate. By integrating, we accumulate all the infinitesimal changes in \( Q \) over the interval from day 0 to day 100 to reconstruct the original amount granted. The outcome of this method yields a function that can predict the remaining grant money for any given day within the span of the grant period.

Initial value problem

In differential equations, an initial value problem is a puzzle where we not only have to solve the equation but also satisfy a specific starting condition. When solving for \( Q(t) \) in our grant disbursement context, we are faced with an initial value problem: \( Q(100) = 0 \), meaning all funds are disbursed after 100 days.

Think of this problem like a detective trying to retrace someone's steps with the knowledge of where they ended up. The 'initial' here is a bit misleading because, in this case, the 'initial' condition is actually provided at the endpoint, \( t=100 \) days. This information is critical because it eliminates the guesswork involved in solving the differential equation – we use it to pin down the precise value of the proportionality constant \( k \), ensuring our function \( Q(t) \) precisely aligns with the real-world scenario of the grant money being fully disbursed by the 100th day.

Proportionality constants

Proportionality constants serve as the bridge between two quantities that vary together in a predictable pattern. In the given problem, \( k \) is the proportionality constant that relates the rate of money disbursement to the square of \(100-t\). The constant \( k \) is what scales the abstract relationship expressed by the differential equation to match the real-life behavior of the federal grant's expenditure.

To illustrate, it's like adjusting the volume on your stereo to achieve the perfect loudness level for your room – \( k \) adjusts the intensity of the disbursement rate to ensure the 2 million dollars is disbursed correctly over the given timeframe. Determining the exact value of \( k \) is a key step in solving our initial value problem, as it allows us to apply the general solution of the differential equation to the specific scenario of our grant. Without \( k \) being accurately calculated, we cannot predict the remaining grant money for any given time.