Noah is going to invest in an account paying an interest rate of compounded continuously. How much would
Noah need to invest, to the nearest hundred dollars, for the value of the account to reach in 10 years?

Question

Noah is going to invest in an account paying an interest rate of

compounded continuously. How much would
Noah need to invest, to the nearest hundred dollars, for the value of the account to reach

in 10 years?

UpStudy AI · Accepted Answer

To determine how much Noah needs to invest today to reach \$12,900 in 10 years with an annual interest rate of $$7\%$$ compounded continuously, we can use the formula for continuous compounding:

$$
A = P e^{rt}
$$

Where:
- $$A$$ is the amount of money accumulated after $$t$$ 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 the money is invested for in years.
- $$e$$ is the base of the natural logarithm, approximately equal to 2.71828.

**Given:**
- $$A = \$12,900$$
- $$r = 0.07$$
- $$t = 10$$ years

**Step 1: Rearrange the Formula to Solve for $$P$$**

$$
P = \frac{A}{e^{rt}} = \frac{12900}{e^{0.07 \times 10}} = \frac{12900}{e^{0.7}}
$$

**Step 2: Calculate $$e^{0.7}$$**

$$
e^{0.7} \approx 2.01375
$$

**Step 3: Compute $$P$$**

$$
P = \frac{12900}{2.01375} \approx 6405.69
$$

**Step 4: Round to the Nearest Hundred Dollars**

$$
P \approx \$6,400
$$

**Conclusion:**
Noah needs to invest approximately **\$6,400** today to reach \$12,900 in 10 years with a 7% continuous compounding interest rate.

**Answer:**  
Noah needs to invest approximately $6,400 today to reach $12,900 in 10 years.
Noah needs to invest approximately \$6,400 today to reach \$12,900 in 10 years with a 7% continuous compounding interest rate.

Noah is going to invest in an account paying an interest rate of compounded continuously. How much would
Noah need to invest, to the nearest hundred dollars, for the value of the account to reach in 10 years?

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Noah is going to invest in an account paying an interest rate of compounded continuously. How much would Noah need to invest, to the nearest hundred dollars, for the value of the account to reach in 10 years?

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Solución

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Noah is going to invest in an account paying an interest rate of compounded continuously. How much would
Noah need to invest, to the nearest hundred dollars, for the value of the account to reach in 10 years?