Question

You currently have $6,500 that you would plan to deposit into an investment account that has an anticipated growth factor of 1.03 per year. Use guess and check to estimate the number of years that it will take for this investment to grow to $10,000. Show your work.​

Answer 1

Using guess and check, we can estimate the number of years it will take for the investment to grow to $10,000 as follows:

First, we'll assume that it will take 5 years for the investment to grow to $10,000.

After 5 years, the investment would grow to:

$6,500 x 1.03^5 = $7,960.35

Since this is still less than $10,000, we need to increase the number of years.

Next, we'll assume that it will take 10 years for the investment to grow to $10,000.

After 10 years, the investment would grow to:

$6,500 x 1.03^10 = $9,020.14

Since this is still less than $10,000, we need to increase the number of years again.

Finally, we'll assume that it will take 15 years for the investment to grow to $10,000.

After 15 years, the investment would grow to:

$6,500 x 1.03^15 = $10,005.58

Since this is slightly more than $10,000, we can estimate that it will take around 15 years for the investment to grow to $10,000.

Therefore, the estimated number of years it will take for the investment to grow to $10,000 is about 15 years.

Answer 2

To find the number of years required for a $6,500 investment to grow to $10,000 with a growth factor of 1.03, we use the compound interest formula and a guess-and-check method. Starting with an estimate and adjusting accordingly, we find that the investment should reach close to $10,000 in approximately 17 years.

Step-by-step explanation:

To estimate the number of years needed for an initial investment of $6,500 to grow to $10,000 with a growth factor of 1.03 per year, we can use the formula for compound interest:

A = P(1 + r)^t

Where:

• A is the amount of money accumulated after n years, including interest.
• P is the principal amount ($6,500).
• r is the annual interest rate (growth factor - 1).
• t is the time in years.

For this problem, we have:

• A = $10,000
• P = $6,500
• r = 0.03 (since the growth factor is 1.03)
• t = unknown

Using the guess and check method:

1. Start with an estimate for t, say t = 10 years.
2. Calculate A = 6500(1 + 0.03)^t to see if it's close to $10,000.
3. Adjust t higher or lower based on the result and calculate again.
4. Repeat step 3 until A is very close to $10,000.

For example, let's start with t = 10 years:

A = 6500(1.03)^10
A = 6500(1.343916379)
A ≈ $8,735.36 (which is less than $10,000, so we increase t)

Next, try t = 20 years:

A = 6500(1.03)^20
A = 6500(1.80611123)
A ≈ $11,739.72 (which is more than $10,000, so we decrease t)

Continue this process until the desired A value is reached. Through various iterations, we can find that the investment reaches close to $10,000 in approximately 17 years.