Problem 50 Austin needs $$\$ 10,000$$ spend... [FREE SOLUTION]

91影视

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

Laurence Hoffmann, Gerald Bradley, David Sobecki, Michael Price

$Math Studyset 91影视 Explanations$ Math

11 Edition

Chapter 3: Problem 50

Austin needs $$\$ 10,000$$ spending money each year, which he takes from his savings account by making $N$ equal withdrawals. Each withdrawal incurs a transaction fee of $$\$ 8$$, and money in his account earns interest at the simple interest rate of $4 \%$. a. The total cost $C$ of managing the account is the transaction cost plus the loss of interest due to withdrawn funds. Express $C$ as a function of $N$. [Hint: You may need the fact that $\left.1+2+\cdots+N=\frac{N(N+1)}{2} .\right]$ b. How many withdrawals should Austin make each year to minimize the total transaction $\operatorname{cost} C(N)$ ?

Short Answer

Expert verified

Austin should make 200 withdrawals each year to minimize the total transaction cost.

Step by step solution

- Define the Variables

Let鈥檚 denote the number of withdrawals Austin makes per year by $N$. Each withdrawal is $$ \$ 8 $$ and incurs a transaction fee.

- Calculate Transaction Fees

The total transaction cost for $N$ withdrawals is calculated as $8N$.

- Calculate the Loss of Interest

Austin withdraws money periodically, reducing the amount of money earning interest in his account.

- Sum of Arithmetic Sequence

The sum of withdrawals can be calculated using the formula for the sum of an arithmetic sequence: $1 + 2 + \cdots + N = \frac{N(N+1)}{2}$.

- Calculate Average Account Balance

The average account balance over the year can be approximated as the midpoint of his decreasing balance due to withdrawals: $ \frac{N(N+1)}{2} $.

- Calculate Interest Loss

The interest rate is $4\%$. Thus, the loss of interest due to withdrawn funds can be expressed as: $0.04 \times \frac{N(N+1)}{2}$.

- Write the Total Cost Function

Combine the transaction fees and the loss of interest to get the total cost: \[ C(N) = 8N + 0.04 \times \frac{N(N+1)}{2} \]. Simplify the equation: \[ C(N) = 8N + 0.02N(N+1) \].

- Differentiate the Cost Function

To find the value of $N$ that minimizes $C(N)$, take the derivative of $C(N)$ with respect to $N$, and set it equal to zero: \[ \frac{dC}{dN} = 8 + 0.04N + 0.02 = 0 \].

- Solve for N

Set the derivative equal to zero and solve for $N$: \[ 8 + 0.04N + 0.02 = 0 \] \[ 0.04N + 0.02 = -8 \] \[ 0.04N = -8 - 0.02 \] \[ N = \frac{-8-0.02}{0.04} \approx 200 \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

simple interest

Simple interest is a straightforward method to calculate the interest earned or paid on a principal amount. It is based on the original principal, the rate of interest, and the time for which the money is invested or borrowed. Mathematically, it is expressed as:
$ \text{Simple Interest} = P \times R \times T $
where:

$P$ is the principal amount
$R$ is the rate of interest
$T$ is the time period

In the problem, Austin's account earns a simple interest rate of $4\%$. This impacts how the loss of interest is calculated when he makes withdrawals throughout the year.

transaction fees

Transaction fees are charges incurred whenever a transaction, such as a withdrawal or transfer, is made. It's essential to account for these fees in cost management for any financial activity. For Austin, each withdrawal incurs a fee of \$8. If he makes $N$ withdrawals in a year, his total transaction cost is calculated as:
$ \text{Total Transaction Cost} = 8N $
Understanding and calculating these fees help in determining the overall cost associated with managing the account.

arithmetic sequence

An arithmetic sequence is a series of numbers in which each term after the first is obtained by adding a constant difference to the previous term. It comes into play in this problem as Austin makes withdrawals spaced evenly throughout the year. The sum of the first $N$ terms of an arithmetic sequence can be calculated using:
\[ 1 + 2 + \text{...} + N = \frac{N(N+1)}{2} \]
This formula helps in determining the total amount withdrawn and the average account balance over the year.

differentiation

Differentiation is a mathematical process used to determine the rate at which a function changes. In cost optimization, differentiation helps us find the minimum or maximum values of a cost function by setting the derivative to zero. For Austin's cost function $C(N)$, we take the derivative:
\[ \frac{dC}{dN} = 8 + 0.04N + 0.02 \]
Setting the derivative equal to zero and solving for $N$ helps us find the number of withdrawals that minimizes the total cost:

cost function

A cost function represents the total cost associated with different levels of an activity. For Austin, the cost function combines transaction fees and the loss of interest due to withdrawals:
\[ C(N) = 8N + 0.04 \times \frac{N(N+1)}{2} \]
Simplifying this gives:
\[ C(N) = 8N + 0.02N(N+1) \]
This function helps in evaluating the overall expense and determining the optimal number of yearly withdrawals.

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