Question

An investor is deciding between two options for a short-term investment. One option has a return \(R\), in dollars, \(t\) months after investment, and is modelled by the equation \(R=100\left(3^t\right)\). The other option has a return \(R\), in dollars, \(t\) months after investment, and is modeled by the equation \(R=350 t\). After 4 months, how much less is the return given by the linear model than the return given by the exponential model? A) \(\$ 1,400\) B) \(\$ 4,050\) C) \(\$ 6,700\) D) \(\$ 8,100\) $$ n-\sqrt{2 n+22}=1 $$

Short answer

The return is \$6,700 less for the linear model after 4 months. So, the correct answer is C) \( \$ 6,700\).

Step-by-step solution

Substitute t in the models

First, we need to find the return values for both models at 4 months (t=4). Let's substitute t=4 in both equations and find the values of R.

Calculate R for exponential model

For the exponential model, the equation is \(R = 100(3^t)\). Plug in t=4: \[R = 100(3^4) = 100(81) = 8100\]

Calculate R for linear model

For the linear model, the equation is \(R = 350t\). Plug in t=4: \[R = 350(4) = 1400\]

Find the difference in returns

Now that we have the returns for both models, we need to find the difference. We are asked to find how much less the return is given by the linear model than by the exponential model. The difference is: \[R_{difference} = R_{exponential} - R_{linear} = 8100 - 1400\]

Calculate the difference and choose the correct option

Calculate the difference in returns: \[R_{difference} = 8100 - 1400 = 6700\] The return is \$6,700 less for the linear model after 4 months. So, the correct answer is C) \( \$ 6,700\).

Key concepts

Understanding Exponential Growth

Exponential growth is a powerful concept in mathematics, often depicting scenarios where quantities increase rapidly over time. This type of growth is characterized by the equation \( R = a(b^t) \), where \( a \) is the initial amount, \( b \) is the growth factor, and \( t \) is the time period.
In exponential models:

• The growth factor \( b \) determines how quickly the increase occurs.
• Any value of \( b \) greater than 1 indicates that the quantity will continue to grow over time.
• Each time period (like months or years) sees the initial amount increased by the power of the growth factor.

In our exercise, the exponential model for return is \( R = 100(3^t) \). Here:

• \( a = 100 \): the initial return in dollars.
• \( b = 3 \): the investment triples each month.
• After 4 months, the return grows to \( 8100 \) dollars.

Breaking Down Linear Equations

Linear equations are foundational in algebra and model scenarios where changes happen at a constant rate. They can be expressed in the form \( R = mt + c \), where \( m \) is the rate of change (or slope), \( t \) is the time variable, and \( c \) is the starting value.
Key points about linear models:

• The rate of change \( m \) is consistent, making the relationship predictable over time.
• They result in a straight-line graph, showing steady progression.
• This model is ideal for situations where growth or decline happens uniformly.

For our linear investment model, \( R = 350t \):

• \( m = 350 \): signifies the monthly increase in the return.
• Since there is no \( c \), the starting value is 0.
• After 4 months, the return is \( 1400 \) dollars, increasing by 350 each month.

Comparing Investment Models

Investment models help us understand and predict how money grows over time in different scenarios. When comparing investment options, we often choose between steady growth and rapid increases.
Important factors in investment models include:

• Return on investment: how much you earn from your initial capital.
• Risk level: how much uncertainty is associated with the investment returns.
• Time frame: the duration over which the investment grows.

In the given problem, we compared two models:

• Exponential model \( R = 100(3^t) \) provided a higher return but comes with greater uncertainty.
• Linear model \( R = 350t \) offered steady, predictable growth but with lower returns.
• After evaluating both models over 4 months, the exponential model yielded an additional \( \$6,700 \) compared to the linear model, highlighting the drastic difference in growth potential.

This comparison illustrates the need to balance growth potential with risk and time when making investment decisions.