Find the monthly payment $$ R $$ needed to have a sinking fund accumulate the future value $$ A $$. The yearly interest rate $$ r $$ and the time $$ t $$ in years are given below. Interest is compounded monthly. $$ A=\$ 6,000 ; r=8 \% ; t=2 $$ $$ R=\$ \square $$ (Round up thathe nearest cent.)

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To find the monthly payment $$ R $$ needed to accumulate a future value $$ A $$ in a sinking fund, we can use the formula for the future value of a sinking fund:

$$
A = R \times \frac{(1 + i)^{nt} - 1}{i}
$$

Where:
- $$ A $$ is the future value of the sinking fund.
- $$ R $$ is the monthly payment.
- $$ i $$ is the monthly interest rate (annual rate divided by 12).
- $$ n $$ is the number of compounding periods per year (12 for monthly).
- $$ t $$ is the number of years.

Given:
- $$ A = 6000 $$
- $$ r = 8\% = 0.08 $$
- $$ t = 2 $$

First, we need to calculate the monthly interest rate $$ i $$ and the total number of payments $$ nt $$:

1. Calculate $$ i $$:
$$
i = \frac{r}{12} = \frac{0.08}{12}
$$

2. Calculate $$ nt $$:
$$
nt = 12 \times t = 12 \times 2
$$

Now, we can substitute these values into the formula and solve for $$ R $$:

$$
6000 = R \times \frac{(1 + i)^{nt} - 1}{i}
$$

Let's calculate $$ i $$ and $$ nt $$ first, and then solve for $$ R $$.
Calculate the value by following steps:
- step0: Calculate:
  $$12\times 2$$
- step1: Multiply the numbers:
  $$24$$
Calculate or simplify the expression $$ 0.08/12 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{0.08}{12}$$
- step1: Convert the expressions:
  $$\frac{\frac{2}{25}}{12}$$
- step2: Multiply by the reciprocal:
  $$\frac{2}{25}\times \frac{1}{12}$$
- step3: Reduce the numbers:
  $$\frac{1}{25}\times \frac{1}{6}$$
- step4: Multiply the fractions:
  $$\frac{1}{25\times 6}$$
- step5: Multiply:
  $$\frac{1}{150}$$
We have calculated the following:

1. The monthly interest rate $$ i $$:
$$
i = \frac{0.08}{12} \approx 0.00666667
$$

2. The total number of payments $$ nt $$:
$$
nt = 12 \times 2 = 24
$$

Now we can substitute $$ i $$ and $$ nt $$ into the future value formula and solve for $$ R $$:

$$
6000 = R \times \frac{(1 + 0.00666667)^{24} - 1}{0.00666667}
$$

Next, we will calculate $$ (1 + 0.00666667)^{24} - 1 $$ and then solve for $$ R $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\left(1+0.00666667\right)^{24}-1$$
- step1: Add the numbers:
  $$1.00666667^{24}-1$$
- step2: Convert the expressions:
  $$\left(\frac{100666667}{100000000}\right)^{24}-1$$
- step3: Rewrite the expression:
  $$\frac{100666667^{24}}{100000000^{24}}-1$$
- step4: Reduce fractions to a common denominator:
  $$\frac{100666667^{24}}{100000000^{24}}-\frac{100000000^{24}}{100000000^{24}}$$
- step5: Transform the expression:
  $$\frac{100666667^{24}-100000000^{24}}{100000000^{24}}$$
We have calculated:

$$
(1 + 0.00666667)^{24} - 1 \approx 0.172888
$$

Now we can substitute this value back into the equation to solve for $$ R $$:

$$
6000 = R \times \frac{0.172888}{0.00666667}
$$

Next, we will calculate $$ \frac{0.172888}{0.00666667} $$ and then solve for $$ R $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{0.172888}{0.00666667}$$
- step1: Convert the expressions:
  $$\frac{\frac{21611}{125000}}{\frac{666667}{100000000}}$$
- step2: Multiply by the reciprocal:
  $$\frac{21611}{125000}\times \frac{100000000}{666667}$$
- step3: Reduce the numbers:
  $$21611\times \frac{800}{666667}$$
- step4: Multiply:
  $$\frac{21611\times 800}{666667}$$
- step5: Multiply:
  $$\frac{17288800}{666667}$$
We have calculated:

$$
\frac{0.172888}{0.00666667} \approx 25.933187
$$

Now we can substitute this value into the equation to solve for $$ R $$:

$$
6000 = R \times 25.933187
$$

To find $$ R $$, we rearrange the equation:

$$
R = \frac{6000}{25.933187}
$$

Now, let's calculate $$ R $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{6000}{25.933187}$$
- step1: Convert the expressions:
  $$\frac{6000}{\frac{25933187}{1000000}}$$
- step2: Multiply by the reciprocal:
  $$6000\times \frac{1000000}{25933187}$$
- step3: Multiply:
  $$\frac{6000\times 1000000}{25933187}$$
- step4: Multiply:
  $$\frac{6000000000}{25933187}$$
We have calculated:

$$
R \approx 231.363773
$$

Rounding this to the nearest cent, we get:

$$
R \approx 231.36
$$

Thus, the monthly payment $$ R $$ needed to accumulate the future value of \$6000 is:

$$
\boxed{231.36}
$$
The monthly payment $$ R $$ is \$231.36.

Find the monthly payment \( R \) needed to have a sinking fund accumulate the future value \( A \). The yearly interest rate \( r \) and the time \( t \) in years are given below. Interest is compounded monthly. \( A=\$ 6,000 ; r=8 \% ; t=2 \) \( R=\$ \square \) (Round up thathe nearest cent.)

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