Questions: Decide whether the following statement makes sense (or is clearly true) or does not make sense (or is clearly false). Explain your reasoning. If interest rates stay at 6% APR and I continue to make my monthly 50 deposits into my retirement plan, I should have at least 40,000 saved when I retire in 35 years. The statement because I will have in my retirement account when I retire in 35 years. (Round to the nearest cent as needed.)

Decide whether the following statement makes sense (or is clearly true) or does not make sense (or is clearly false). Explain your reasoning.
If interest rates stay at 6% APR and I continue to make my monthly 50 deposits into my retirement plan, I should have at least 40,000 saved when I retire in 35 years.

The statement because I will have in my retirement account when I retire in 35 years.
(Round to the nearest cent as needed.)

Transcript text: Decide whether the following statement makes sense (or is clearly true) or does not make sense (or is clearly false). Explain your reasoning. If interest rates stay at $6 \%$ APR and I continue to make my monthly $\$ 50$ deposits into my retirement plan, I should have at least $\$ 40,000$ saved when I retire in 35 years. The statement $\square$ because I will have $\$$ $\square$ in my retirement account when I retire in 35 years. (Round to the nearest cent as needed.)

Solution

Solution Steps

To determine whether the statement makes sense, we need to calculate the future value of a series of monthly deposits into a retirement account with a given annual interest rate. We will use the future value of an annuity formula for this purpose.

Step 1: Given Values

We have the following parameters for the retirement plan:

Monthly deposit: $ P = 50 $
Annual interest rate: $ r = 0.06 $
Number of years: $ t = 35 $

Step 2: Calculate Total Months

The total number of months over which deposits will be made is: \[ n = t \times 12 = 35 \times 12 = 420 \]

Step 3: Calculate Monthly Interest Rate

The monthly interest rate is calculated as: \[ i = \frac{r}{12} = \frac{0.06}{12} = 0.005 \]

Step 4: Future Value of Annuity

The future value $ FV $ of the annuity (the total amount saved at retirement) can be calculated using the formula: \[ FV = P \times \left( \frac{(1 + i)^n - 1}{i} \right) \] Substituting the values: \[ FV = 50 \times \left( \frac{(1 + 0.005)^{420} - 1}{0.005} \right) \]

Step 5: Calculate Future Value

After performing the calculations, we find: \[ FV \approx 71235.51 \]

Step 6: Evaluate the Statement

The statement claims that the total savings will be at least $ 40000 $. Since: \[ 71235.51 \geq 40000 \] the statement is true.