Problem 58
Question
The value of a recently issued General Electric bond increases in value at the rate of \(40 e^{0.04 t}\) dollars per year, where \(t=0\) represents \(2013 .\) a. Find a formula for the total increase in the value of the stock within \(t\) years of \(2013 .\) b. Use your formula to find the total increase from 2013 to 2028 .
Step-by-Step Solution
Verified Answer
The total increase from 2013 to 2028 is approximately $822.10.
1Step 1: Understand the given rate of increase
The given rate of increase in the value of the bond is expressed as a function of time: \( \frac{dV}{dt} = 40 e^{0.04t} \). This indicates that the value increases at this exponential rate each year.
2Step 2: Set up the integral for total increase
To find the total increase in value over \( t \) years, we need to integrate the rate of increase with respect to time from 0 to \( t \). The integral is given by:\[ V(t) - V(0) = \int_{0}^{t} 40 e^{0.04t} \, dt \]
3Step 3: Evaluate the integral
Calculate the integral:\[ \int 40 e^{0.04t} \, dt = \frac{40}{0.04} e^{0.04t} = 1000 e^{0.04t} \]Thus, the definite integral from 0 to \( t \) is:\[ V(t) - V(0) = \left[ 1000 e^{0.04t} \right]_0^t = 1000 e^{0.04t} - 1000 \]
4Step 4: Write the formula for total increase
Subtracting the initial value simplifies it to:\[ V(t) = C + 1000(e^{0.04t} - 1) \] Here, \( C \) is the initial value, so the total increase is:\[ \Delta V(t) = 1000(e^{0.04t} - 1) \]
5Step 5: Calculate the total increase from 2013 to 2028
To find the increase from 2013 to 2028, substitute \( t = 15 \) (as 2028-2013=15) into the formula:\[ \Delta V(15) = 1000(e^{0.04 imes 15} - 1) \]Calculate the exponential term:\( 0.04 \times 15 = 0.6 \) and \( e^{0.6} \approx 1.8221 \), then the total increase:\[ \Delta V(15) = 1000(1.8221 - 1) = 1000(0.8221) = 822.1 \]
Key Concepts
Exponential GrowthDefinite IntegralRate of Change
Exponential Growth
Exponential growth describes how a quantity increases rapidly over time.In mathematical terms, an exponential growth pattern is characterized by the formula \( A = A_0 e^{kt} \).Here, \( A_0 \) is the initial amount, \( k \) is the growth rate, and \( t \) is the time period.
For the problem at hand, the rate at which the bond value increases is given by \( 40 e^{0.04t} \).This means initially, the bond increases by 40 dollars per year but as time passes, the increase becomes significantly larger.This is because the term \( e^{0.04t} \) continuously expands, causing compounded growth.
Here's why it matters:- **Doubling Time**: With exponential growth, you can calculate how fast a quantity doubles.- **Continuous Growth**: Factors like inflation or interest often compound continuously, akin to this pattern.Understanding this concept is crucial since many natural and economic processes follow similar exponential laws.
For the problem at hand, the rate at which the bond value increases is given by \( 40 e^{0.04t} \).This means initially, the bond increases by 40 dollars per year but as time passes, the increase becomes significantly larger.This is because the term \( e^{0.04t} \) continuously expands, causing compounded growth.
Here's why it matters:- **Doubling Time**: With exponential growth, you can calculate how fast a quantity doubles.- **Continuous Growth**: Factors like inflation or interest often compound continuously, akin to this pattern.Understanding this concept is crucial since many natural and economic processes follow similar exponential laws.
Definite Integral
The definite integral is a fundamental tool in calculus used to calculate the total accumulation of a quantity.In this exercise, it helps determine the total increase in the bond's value over time.
Mathematically, the definite integral sums up the rate of change over a particular interval.The integral \( \int_{0}^{t} 40 e^{0.04t} \, dt \) calculates the total change in value from year 0 to year \( t \).
Some key points regarding definite integrals are:- **Total Accumulation**: It provides the total value added from the starting point.- **Area Under the Curve**: In a graphical sense, the integral represents the area beneath the curve of the given function.
By solving the integral \( 40 e^{0.04t} \), you get a formula for the total increase.From Step 3, this evaluates to \( 1000(e^{0.04t} - 1) \), symbolizing how much the bond value has grown from its initial worth.
Mathematically, the definite integral sums up the rate of change over a particular interval.The integral \( \int_{0}^{t} 40 e^{0.04t} \, dt \) calculates the total change in value from year 0 to year \( t \).
Some key points regarding definite integrals are:- **Total Accumulation**: It provides the total value added from the starting point.- **Area Under the Curve**: In a graphical sense, the integral represents the area beneath the curve of the given function.
By solving the integral \( 40 e^{0.04t} \), you get a formula for the total increase.From Step 3, this evaluates to \( 1000(e^{0.04t} - 1) \), symbolizing how much the bond value has grown from its initial worth.
Rate of Change
The rate of change tells us how a quantity varies over time.In calculus, this is often represented by the derivative \( \frac{dV}{dt} \).In our problem, the function \( \frac{dV}{dt} = 40 e^{0.04t} \) describes how quickly the bond value changes each year.
Here's why understanding rate of change matters:- **Instantaneous vs. Average Rate**: Unlike average rates, derivatives give the rate at a precise moment.- **Forecasting**: Knowing the rate helps predict future trends and behaviors.
With this exercise, the exponential formula shows that as \( t \) increases, the rate \( 40 e^{0.04t} \) grows.This rapid increase highlights exponential properties, as the rate itself is not constant but accelerates over time.Thus, mastering this concept illuminates how rapidly changing systems, like financial markets, function.
Here's why understanding rate of change matters:- **Instantaneous vs. Average Rate**: Unlike average rates, derivatives give the rate at a precise moment.- **Forecasting**: Knowing the rate helps predict future trends and behaviors.
With this exercise, the exponential formula shows that as \( t \) increases, the rate \( 40 e^{0.04t} \) grows.This rapid increase highlights exponential properties, as the rate itself is not constant but accelerates over time.Thus, mastering this concept illuminates how rapidly changing systems, like financial markets, function.
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