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. $$ A=\$ 4,000 ; r=2.5 \% ; \mathrm{t}=1 $$ $$ \mathrm{R}=\$ \square $$ (Round up to hie 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 series of payments (ordinary annuity) compounded at a certain interest rate. The formula is given by:

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

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.

### Step 1: Extract known values
- $$ A = 4000 $$
- $$ r = 2.5\% = 0.025 $$
- $$ t = 1 $$

### Step 2: Calculate the monthly interest rate
$$
i = \frac{r}{12} = \frac{0.025}{12}
$$

### Step 3: Calculate the number of compounding periods
$$
n = 12 \quad (\text{monthly compounding})
$$
$$
nt = 12 \times 1 = 12
$$

### Step 4: Substitute values into the formula and solve for $$ R $$
Rearranging the formula to solve for $$ R $$:
$$
R = \frac{A \cdot i}{(1 + i)^{nt} - 1}
$$

Now, we will calculate $$ i $$ and then substitute all values into the formula to find $$ R $$.

Let's perform the calculations.

1. Calculate $$ i $$:
$$
i = \frac{0.025}{12}
$$

2. Substitute into the formula for $$ R $$:
$$
R = \frac{4000 \cdot i}{(1 + i)^{12} - 1}
$$

Now, I will calculate these values.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{0.025}{12}$$
- step1: Convert the expressions:
  $$\frac{\frac{1}{40}}{12}$$
- step2: Multiply by the reciprocal:
  $$\frac{1}{40}\times \frac{1}{12}$$
- step3: Multiply the fractions:
  $$\frac{1}{40\times 12}$$
- step4: Multiply:
  $$\frac{1}{480}$$
Calculate or simplify the expression $$ 4000*(0.025/12)/((1+(0.025/12))^12-1) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{4000\left(\frac{0.025}{12}\right)}{\left(\left(1+\left(\frac{0.025}{12}\right)\right)^{12}-1\right)}$$
- step1: Remove the parentheses:
  $$\frac{4000\left(\frac{0.025}{12}\right)}{\left(1+\left(\frac{0.025}{12}\right)\right)^{12}-1}$$
- step2: Divide the terms:
  $$\frac{4000\left(\frac{0.025}{12}\right)}{\left(1+\frac{1}{480}\right)^{12}-1}$$
- step3: Add the numbers:
  $$\frac{4000\left(\frac{0.025}{12}\right)}{\left(\frac{481}{480}\right)^{12}-1}$$
- step4: Divide the terms:
  $$\frac{4000\times \frac{1}{480}}{\left(\frac{481}{480}\right)^{12}-1}$$
- step5: Multiply the numbers:
  $$\frac{\frac{25}{3}}{\left(\frac{481}{480}\right)^{12}-1}$$
- step6: Subtract the numbers:
  $$\frac{\frac{25}{3}}{\frac{481^{12}-480^{12}}{480^{12}}}$$
- step7: Multiply by the reciprocal:
  $$\frac{25}{3}\times \frac{480^{12}}{481^{12}-480^{12}}$$
- step8: Rewrite the expression:
  $$\frac{25}{3}\times \frac{3^{12}\times 160^{12}}{481^{12}-480^{12}}$$
- step9: Reduce the numbers:
  $$25\times \frac{3^{11}\times 160^{12}}{481^{12}-480^{12}}$$
- step10: Multiply:
  $$\frac{25\times 3^{11}\times 160^{12}}{481^{12}-480^{12}}$$
### Step 1: Calculate the monthly interest rate $$ i $$
$$
i = \frac{0.025}{12} \approx 0.00208333
$$

### Step 2: Substitute into the formula for $$ R $$
Now we substitute $$ i $$ into the formula for $$ R $$:
$$
R = \frac{4000 \cdot 0.00208333}{(1 + 0.00208333)^{12} - 1}
$$

Calculating $$ (1 + i)^{12} $$:
$$
(1 + 0.00208333)^{12} \approx 1.0254
$$

Now substituting this back into the formula for $$ R $$:
$$
R = \frac{4000 \cdot 0.00208333}{1.0254 - 1}
$$
$$
R = \frac{4000 \cdot 0.00208333}{0.0254}
$$

Now, let's calculate $$ R $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{4000\times 0.00208333}{0.0254}$$
- step1: Multiply the numbers:
  $$\frac{8.33332}{0.0254}$$
- step2: Convert the expressions:
  $$\frac{\frac{208333}{25000}}{\frac{127}{5000}}$$
- step3: Multiply by the reciprocal:
  $$\frac{208333}{25000}\times \frac{5000}{127}$$
- step4: Reduce the numbers:
  $$\frac{208333}{5}\times \frac{1}{127}$$
- step5: Multiply the fractions:
  $$\frac{208333}{5\times 127}$$
- step6: Multiply:
  $$\frac{208333}{635}$$
### Final Calculation for $$ R $$
After performing the calculations, we find:
$$
R \approx 328.083465
$$

### Step 3: Round up to the nearest cent
Rounding $$ R $$ to the nearest cent gives:
$$
R \approx 328.08
$$

Thus, the monthly payment $$ R $$ needed to accumulate the future value $$ A $$ is:
$$
\boxed{328.08}
$$
The monthly payment $$ R $$ needed is \$328.08.

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. \( A=\$ 4,000 ; r=2.5 \% ; \mathrm{t}=1 \) \( \mathrm{R}=\$ \square \) (Round up to hie nearest cent.)

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