If Tanisha has ​$1000 to invest at 8​% per annum compounded monthly​, how long will it be before she has ​$1750​? If the compounding is​ continuous, how long will it​ be?
Compounding monthly​, it will be about _ $ years before Tanisha has ​$1750.
​(Round to two decimal places as​ needed.)

To solve this problem, we need to find the time it takes for an investment to grow from $1000 to $1750 with two different compounding methods: monthly and continuous.

### 1. Compounded Monthly
For monthly compounding, the formula is:
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
where:
- $ A $ is the future amount ($1750),
- $ P $ is the principal amount ($1000),
- $ r $ is the annual interest rate (8%, or 0.08 as a decimal),
- $ n $ is the number of times the interest is compounded per year (12 for monthly),
- $ t $ is the time in years.

#### Step-by-Step Solution
1. Substitute the values into the formula:
   \[
   1750 = 1000 \left(1 + \frac{0.08}{12}\right)^{12t}
   \]
2. Divide both sides by 1000:
   \[
   1.75 = \left(1 + \frac{0.08}{12}\right)^{12t}
   \]
3. Simplify the expression inside the parentheses:
   \[
   1.75 = \left(1 + 0.0066667\right)^{12t} \approx (1.006667)^{12t}
   \]
4. Take the natural logarithm of both sides to solve for $ t $:
   \[
   \ln(1.75) = 12t \cdot \ln(1.006667)
   \]
5. Solve for $ t $:
   \[
   t = \frac{\ln(1.75)}{12 \cdot \ln(1.006667)} \approx 6.80 \text{ years}
   \]

So, with monthly compounding, it will take approximately **6.80 years** for Tanisha's investment to reach $1750.

### 2. Continuous Compounding
For continuous compounding, the formula is:
\[
A = Pe^{rt}
\]
where:
- $ A = 1750 $,
- $ P = 1000 $,
- $ r = 0.08 $,
- $ e $ is the base of the natural logarithm (approximately 2.71828).

#### Step-by-Step Solution
1. Substitute the values into the formula:
   \[
   1750 = 1000 \cdot e^{0.08t}
   \]
2. Divide both sides by 1000:
   \[
   1.75 = e^{0.08t}
   \]
3. Take the natural logarithm of both sides to solve for $ t $:
   \[
   \ln(1.75) = 0.08t
   \]
4. Solve for $ t $:
   \[
   t = \frac{\ln(1.75)}{0.08} \approx 7.31 \text{ years}
   \]

With continuous compounding, it will take approximately **7.31 years** for the investment to grow to $1750.

### Final Answers
- **Compounded Monthly**: About 6.80 years
- **Compounded Continuously**: About 7.31 years

Would you like further details on any step?

Here are some related questions to explore:
1. How does increasing the interest rate affect the time required to reach the same amount?
2. What would happen if the principal amount were doubled?
3. How does the compounding frequency impact the time required for growth?
4. What is the difference in growth over 10 years with monthly versus continuous compounding?
5. If the investment were instead compounded quarterly, how long would it take?

**Tip:** Continuous compounding is ideal for maximizing interest over time since it assumes constant reinvestment.

If Tanisha has $1000 to invest at 8% per annum compounded monthly, how long will it be before she has $1750? If the compounding is continuous, how long will it be?

Learn how long it takes for a $1000 investment to grow to $1750 with 8% interest, using both monthly and continuous compounding. This problem covers key concepts like exponential growth and compound interest, providing a step-by-step solution. Suitable for students in Grades 10-12.

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