Problem 56 - College Algebra and Calculus: An Applied Approach

Question

Cash Flow The rate of disbursement \(d Q / d t\) of a \(\$ 4\) million federal grant is proportional to the square of \(100-t\), where \(t\) is the time (in days, \(0 \leq t \leq 100\) ) and \(Q\) is the amount that remains to be disbursed. Find the amount that remains to be disbursed after 50 days. Assume that the entire grant will be disbursed after 100 days.

Step-by-Step Solution

Answer

To calculate Q(50), we need to integrate the given differential equation and apply the boundary condition to find the constant of proportionality. The exact value will be determined based on the result of the integration.

Step 1: Write Down the Differential Equation

The disbursement rate \(d Q / d t\) is proportional to the square of \(100-t\), so we can write this as: \(d Q / dt = k \cdot (100-t)^2\), where \(k\) is the constant of proportionality.

Step 2: Integrate the Differential Equation

Integrating both sides with respect to \(t\), we obtain: \(\int d Q = k \int (100-t)^2 dt\). The solution to this integral gives us the function \(Q(t)\), which describes the amount that remains to be disbursed at time \(t\). We need to find the constant \(k\) by the condition that the entire grant will be disbursed after 100 days, i.e, \(Q(100) = 0\).

Step 3: Apply the Boundary Condition

Substitute \(Q(100) = 0\) into the equation obtained from the integration, we can solve for \(k\).

Step 4: Solve for Q(50)

Substitute \(t = 50\) into the equation for \(Q(t)\) after we have obtained \(k\), we can calculate the exact amount that remains to be disbursed after 50 days.

Key Concepts

Rate of Disbursement

In the context of a federal grant, the **rate of disbursement** refers to how quickly the funds are allocated or used over time. Here, we're dealing with a differential equation that models this rate: \( \frac{dQ}{dt} = k \cdot (100-t)^2 \). This shows the relationship between the rate at which funds are disbursed \( \frac{dQ}{dt} \) and the remaining time.
The expression \((100-t)^2\) illustrates that the rate is not constant but changes with time. It's important because:

• It shows how disbursement slows down as time approaches 100 days.
• The square indicates how drastically the rate can change over time.

As you can see, understanding the rate of disbursement is crucial for predicting how funds are managed and spent.

Federal Grant

A **federal grant** is a sum of money given by the government for a specific purpose. In this scenario, the grant is $4 million, intended to be fully disbursed within a time frame of 100 days. This necessitates careful calculation to ensure the funds are utilized effectively and entirely by the deadline.
The problem highlights:

• The importance of accurate planning and allocation.
• How mathematics, particularly differential equations, can be used in financial planning.

A federal grant supports various initiatives, and factors like time and rate of disbursement help in strategizing its use.

Boundary Conditions

**Boundary conditions** are essential in solving differential equations because they provide specific points that help determine unknown constants. Here, the condition \( Q(100) = 0 \) tells us that all funds will be disbursed by day 100.
Knowing this, we can:

• Compute the value of the constant \( k \).
• Develop an exact solution for \( Q(t) \), the remaining funds at given times.

Boundary conditions transform a general solution to a specific one, tailored to real-world scenarios like managing a grant.

Integration

**Integration** is a fundamental step in solving differential equations. It involves finding the integral of the equation, turning a rate of change (like disbursement rate) into a total amount function.
Here, we integrate: \[\int d Q = k \int (100-t)^2 dt\] This translates the disbursement rate into a function, \( Q(t) \), representing the amount left to disburse.
Integration is key because:

• It bridges the gap between rates and totals.
• It helps derive solutions based on real-world data, such as how much funds remain.

This process illustrates the power of calculus in practical applications.