Select An Amount Of Money You'd Like To Invest

Question

Select An Amount Of Money That You Would Like To Invest For Example Select an amount of money that you would like to invest (for example $1000.00). This will be your P₀ value. Let your interest rate be k = 0.5%. Write out the exponential function using the P₀ and k values you have. Determine the value of your investment after 1, 5, and 10 years. Now, find the doubling time T for your investment. In other words, at what time would your initial deposit double in value? Repeat steps 3 through 5 for k = 1%. Repeat steps 3 through 5 for k = 1.5%. work out the details clearly

Dr. Jack HW Helper · Accepted Answer

Introduction Investing money wisely requires understanding how compounded interest influences growth over time. This paper explores exponential growth models for investments at varying interest rates, calculating the value of an initial deposit over specific periods and determining the time required for the investment to double at different rates. Investment Model and Exponential Function The exponential growth of an investment can be modeled using the formula: $$ P(t) = P_0 \times e^{k t} $$ where: $ P(t) $ is the amount of money after time $ t $ (in years), $ P_0 $ is the initial principal amount, $ k $ is the interest rate expressed as a decimal, $ e $ is Euler’s number, approximately 2.71828. Given: - Initial principal $ P_0 = 1000 $ dollars, - Interest rate $ k = 0.5\% = 0.005 $. The exponential function becomes: $$ P(t) = 1000 \times e^{0.005 t} $$ For the other interest rates, the functions are: - For $ k = 1\% = 0.01 $: $$ P(t) = 1000 \times e^{0.01 t} $$ - For $ k = 1.5\% = 0.015 $: $$ P(t) = 1000 \times e^{0.015 t} $$ Calculations at Different Time Periods Calculating the investment values at 1, 5, and 10 years. 1. For $ k = 0.005 $: - After 1 year: $$ P(1) = 1000 \times e^{0.005 \times 1} = 1000 \times e^{0.005} \approx 1000 \times 1.0050125 \approx \$1005.01 $$ - After 5 years: $$ P(5) = 1000 \times e^{0.005 \times 5} = 1000 \times e^{0.025} \approx 1000 \times 1.025315 \approx \$1025.32 $$ - After 10 years: $$ P(10) = 1000 \times e^{0.005 \times 10} = 1000 \times e^{0.05} \approx 1000 \times 1.051271 \approx \$1051.27 $$ 2. For $ k = 0.01 $: - After 1 year: $$ P(1) = 1000 \times e^{0.01} \approx 1000 \times 1.010050 \approx \$1010.05 $$ - After 5 years: $$ P(5) = 1000 \times e^{0.05} \approx 1000 \times 1.051271 \approx \$1051.27 $$ - After 10 years: $$ P(10) = 1000 \times e^{0.1} \approx 1000 \times 1.105171 \approx \$1105.17 $$ 3. For $ k = 0.015 $: - After 1 year: \[ P(1) = 1000 \times e^{0.015} \approx 1000 \times 1.015113 \a

Select An Amount Of Money You'd Like To Invest

Select An Amount Of Money That You Would Like To Invest For Example

Paper For Above instruction

Introduction

Investment Model and Exponential Function

Calculations at Different Time Periods

Calculating Doubling Time T

Discussion

Conclusion

References