Bobby and Shelley deposit $\$800.00$ into a savings account which earns $6\%$ interest compounded continuously. They want to use the money in the account to go on a trip in $3$ years. How much will they be able to spend? $\newline$ Use the formula $A = Pe^{rt}$ , where $A$ is the balance (final amount), $P$ is the principal (starting amount), $e$ is the base of natural logarithms ( $\approx 2.71828$ ), $r$ is the interest rate expressed as a decimal, and $t$ is the time in years. $\newline$ Round your answer to the nearest cent. $\newline$ $\$$ ____

Full solution

Q. Bobby and Shelley deposit

\$800.00

into a savings account which earns

6\%

interest compounded continuously. They want to use the money in the account to go on a trip in

3

years. How much will they be able to spend?

\newline

Use the formula

A = Pe^{rt}

, where

A

is the balance (final amount),

P

is the principal (starting amount),

e

is the base of natural logarithms (

\approx 2.71828

r

is the interest rate expressed as a decimal, and

t

is the time in years.

\newline

Round your answer to the nearest cent.

\newline

\$

____

Identify values for P, r, and t: Identify the values for P, r, and t. $\newline$ Principal amount $P$ = $\$800$ $\newline$ Rate of interest $r$ = $6\%$ or $0.06$ when expressed as a decimal $\newline$ Time in years $t$ = $3$ years
Calculate final amount using continuous compounding interest formula: Use the continuous compounding interest formula $A = Pe^{rt}$ to calculate the final amount. $\newline$ Substitute $P = 800$ , $r = 0.06$ , and $t = 3$ into the formula. $\newline$ $A = 800 \times e^{(0.06 \times 3)}$
Calculate exponent part of the formula: Calculate the exponent part of the formula. $0.06 \times 3 = 0.18$
Calculate $e$ raised to the power of $0.18$ : Calculate $e$ raised to the power of $0.18$ . $e^{0.18} \approx 2.71828^{0.18} \approx 1.1972173...$
Multiply principal amount by $e^{0.18}$ to find final amount: Multiply the principal amount by the value of $e$ raised to the power of $0.18$ to find the final amount. $\newline$ $A = 800 \times 1.1972173...$ $\newline$ $A \approx 800 \times 1.1972173 \approx 957.77384$
Round final amount to nearest cent: Round the final amount to the nearest cent. $A \approx \$957.77$