Math  /  Algebra

QuestionUse the formula for continuous compounding to compute the balance in the account after 1,5, and 20 years. Also, find the APY for the account. A $3000\$ 3000 deposit in an account with an APR of 9.5%9.5 \%
The balance in the account after 1 year is approximately $\$ \square (Round to the nearest cent as needed.)

Studdy Solution

STEP 1

1. The initial deposit amount is P=$3000P = \$3000.
2. The annual percentage rate (APR) is r=9.5%=0.095r = 9.5\% = 0.095.
3. The formula for continuous compounding is given by A=PertA = Pe^{rt}, where AA is the amount after time tt, PP is the principal, rr is the annual interest rate, and tt is the time in years.
4. We need to compute the balance after 1, 5, and 20 years.
5. The annual percentage yield (APY) can be found using the formula APY=er1\text{APY} = e^r - 1.

STEP 2

1. Calculate the balance in the account after 1 year.
2. Calculate the balance in the account after 5 years.
3. Calculate the balance in the account after 20 years.
4. Calculate the APY for the account.

STEP 3

Use the continuous compounding formula A=PertA = Pe^{rt} to calculate the balance after 1 year.
A=3000×e0.095×1 A = 3000 \times e^{0.095 \times 1}

STEP 4

Calculate the value of e0.095e^{0.095}.
e0.0951.0995 e^{0.095} \approx 1.0995

STEP 5

Multiply the principal by e0.095e^{0.095} to find the balance after 1 year.
A3000×1.0995=3298.50 A \approx 3000 \times 1.0995 = 3298.50

STEP 6

Use the continuous compounding formula A=PertA = Pe^{rt} to calculate the balance after 5 years.
A=3000×e0.095×5 A = 3000 \times e^{0.095 \times 5}

STEP 7

Calculate the value of e0.475e^{0.475}.
e0.4751.6084 e^{0.475} \approx 1.6084

STEP 8

Multiply the principal by e0.475e^{0.475} to find the balance after 5 years.
A3000×1.6084=4825.20 A \approx 3000 \times 1.6084 = 4825.20

STEP 9

Use the continuous compounding formula A=PertA = Pe^{rt} to calculate the balance after 20 years.
A=3000×e0.095×20 A = 3000 \times e^{0.095 \times 20}

STEP 10

Calculate the value of e1.9e^{1.9}.
e1.96.6859 e^{1.9} \approx 6.6859

STEP 11

Multiply the principal by e1.9e^{1.9} to find the balance after 20 years.
A3000×6.6859=20057.70 A \approx 3000 \times 6.6859 = 20057.70

STEP 12

Calculate the APY using the formula APY=er1\text{APY} = e^r - 1.
APY=e0.0951 \text{APY} = e^{0.095} - 1

STEP 13

Calculate the value of e0.095e^{0.095} again.
e0.0951.0995 e^{0.095} \approx 1.0995

STEP 14

Subtract 1 from e0.095e^{0.095} to find the APY.
APY1.09951=0.09959.95% \text{APY} \approx 1.0995 - 1 = 0.0995 \approx 9.95\%
Solution: - The balance after 1 year is approximately $3298.50\$3298.50. - The balance after 5 years is approximately $4825.20\$4825.20. - The balance after 20 years is approximately $20057.70\$20057.70. - The APY for the account is approximately 9.95%9.95\%.

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