The cost of painting a wall increases by a fixed percentage each year. In 1970, the cost was $2,000, and in 1979, it was $3,600. What was the cost of painting in 1988?

Options

A. 1111
B. 2111
C. 3600
D. 6480

Answer

To solve the problem step by step, we will use the concept of compound interest since the cost of painting increases by a fixed percentage each year.

### Step 1: Identify the initial values
- Initial cost in 1970 = $2000
- Cost in 1979 = $3600

### Step 2: Determine the time period
- From 1970 to 1979, the time period is 9 years.

### Step 3: Use the compound interest formula
The formula for compound interest is:
\[ A = P \left(1 + \frac{r}{100}\right)^t \]
Where:
- \(A\) = Amount after time \(t\)
- \(P\) = Principal amount (initial cost)
- \(r\) = Rate of interest (percentage increase per year)
- \(t\) = Time in years

In our case:
- \(A = 3600\)
- \(P = 2000\)
- \(t = 9\)

### Step 4: Set up the equation
Substituting the known values into the formula:
\[ 3600 = 2000 \left(1 + \frac{r}{100}\right)^9 \]

### Step 5: Simplify the equation
Divide both sides by 2000:
\[ 1.8 = \left(1 + \frac{r}{100}\right)^9 \]

### Step 6: Take the ninth root
To solve for \(1 + \frac{r}{100}\), take the ninth root of both sides:
\[ 1 + \frac{r}{100} = 1.8^{\frac{1}{9}} \]

### Step 7: Calculate \(1.8^{\frac{1}{9}}\)
Using a calculator, we find:
\[ 1 + \frac{r}{100} \approx 1.083 \]
So,
\[ \frac{r}{100} \approx 0.083 \quad \Rightarrow \quad r \approx 8.3\% \]

### Step 8: Calculate the cost in 1988
Now, we need to find the cost in 1988, which is 9 years after 1979. So, we will use the cost in 1979 as the new principal:
\[ A = 3600 \left(1 + \frac{r}{100}\right)^9 \]
Substituting \(r \approx 8.3\):
\[ A = 3600 \left(1.083\right)^9 \]

### Step 9: Calculate \(A\)
Using a calculator:
\[ A \approx 3600 \times 1.834 \approx 6602.4 \]

### Step 10: Round to the nearest dollar
Thus, the cost of painting in 1988 is approximately:
\[ \text{Cost in 1988} \approx 6602 \]

### Final Answer
The cost of painting a wall in 1988 was approximately **$6602**.

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