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Tim plans to invest money into an account that earns 4.5% and is compounded continuously. How much should he invest so that he has $10,000 after 10 years? (Round to the nearest penny.)

Tim plans to invest money into an account that earns 4.5% and is compounded continuously. How much should he invest so that he has $10,000 after 10 years? (Round to the nearest penny.)

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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**Investment Problem: Continuous Compounding Interest** **Problem Statement:** Tim plans to invest money into an account that earns 4.5% interest and is compounded continuously. How much should he invest so that he has $10,000 after 10 years? (Round to the nearest penny.) **Explanation:** To solve this problem, we'll use the formula for continuous compounding interest: \[ A = Pe^{rt} \] Where: - \( A \) is the amount of money accumulated after n years, including interest. - \( P \) is the principal amount (the initial amount of money). - \( r \) is the annual interest rate (in decimal). - \( t \) is the time in years. - \( e \) is the base of the natural logarithm, approximately equal to 2.71828. Given: - \( A = 10,000 \) - \( r = 0.045 \) - \( t = 10 \) The goal is to solve for \( P \): \[ 10,000 = Pe^{0.045 \times 10} \] \[ 10,000 = Pe^{0.45} \] To find \( P \): 1. Calculate \( e^{0.45} \). 2. Divide both sides by the result from step 1 to isolate \( P \). Calculate \( P \) to determine how much Tim should invest initially.
Transcribed Image Text:**Investment Problem: Continuous Compounding Interest** **Problem Statement:** Tim plans to invest money into an account that earns 4.5% interest and is compounded continuously. How much should he invest so that he has $10,000 after 10 years? (Round to the nearest penny.) **Explanation:** To solve this problem, we'll use the formula for continuous compounding interest: \[ A = Pe^{rt} \] Where: - \( A \) is the amount of money accumulated after n years, including interest. - \( P \) is the principal amount (the initial amount of money). - \( r \) is the annual interest rate (in decimal). - \( t \) is the time in years. - \( e \) is the base of the natural logarithm, approximately equal to 2.71828. Given: - \( A = 10,000 \) - \( r = 0.045 \) - \( t = 10 \) The goal is to solve for \( P \): \[ 10,000 = Pe^{0.045 \times 10} \] \[ 10,000 = Pe^{0.45} \] To find \( P \): 1. Calculate \( e^{0.45} \). 2. Divide both sides by the result from step 1 to isolate \( P \). Calculate \( P \) to determine how much Tim should invest initially.
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