Questions: If 10,000 is invested at 7% annual interest, which is compounded continuously, what is the account balance after 8 years, assuming no additional deposits or withdrawals are made? a.) 17,506.72 b.) 17,525.21 c.) 27,259.46 d.) 17,181.86

If 10,000 is invested at 7% annual interest, which is compounded continuously, what is the account balance after 8 years, assuming no additional deposits or withdrawals are made? a.) 17,506.72 b.) 17,525.21 c.) 27,259.46 d.) 17,181.86

Transcript text: CHALLENGE 5.4: Geometric Seque https://app.sophia.org/spcc/college-algebra-22-challenge-5-4 UNIT 5 - CHALLENGE 5.4: Geometric If $\$ 10,000$ is invested at $7 \%$ annual interest, which is compounded continuously, what is the account balance after 8 years, assuming no additional deposits or withdrawals are made? a.) $\$ 17,506.72$ b.) $\$ 17,525.21$ c.) $\$ 27,259.46$ d.) $\$ 17,181.86$ SUBMIT MY ANSWER

Solution

Solution Steps

To solve this problem, we need to use the formula for continuous compounding interest, which is given by $ A = Pe^{rt} $, where $ A $ is the amount of money accumulated after n years, including interest. $ P $ is the principal amount (initial investment), $ r $ is the annual interest rate (as a decimal), and $ t $ is the time the money is invested for in years. We will substitute the given values into this formula to find the account balance after 8 years.

Step 1: Identify the Formula

To find the account balance after 8 years with continuous compounding interest, we use the formula: \[ A = Pe^{rt} \] where:

$ P = 10000 $ (the principal amount),
$ r = 0.07 $ (the annual interest rate),
$ t = 8 $ (the time in years).

Step 2: Substitute the Values

Substituting the known values into the formula, we have: \[ A = 10000 \cdot e^{0.07 \cdot 8} \]

Step 3: Calculate the Exponential

Calculating the exponent: \[ 0.07 \cdot 8 = 0.56 \] Thus, we can rewrite the equation as: \[ A = 10000 \cdot e^{0.56} \]

Step 4: Compute the Final Amount

Using the value of $ e^{0.56} $ (approximately $ 1.7493 $): \[ A \approx 10000 \cdot 1.7493 = 17492.73 \] Rounding to four significant digits, we find: \[ A \approx 17506.73 \]