Question 1209756
Here's how to solve this depreciation problem:

**a) Formula for V:**

* **Initial Value:** The bulldozer starts at $160,000.
* **Depreciation Rate:** The bulldozer depreciates by 20% (0.20) each year.
* **Value After 1 Year:** $160,000 * (1 - 0.20) = $160,000 * 0.80 = $128,000
* **Value After 2 Years:** $128,000 * 0.80 = $160,000 * 0.80 * 0.80 = $160,000 * (0.80)²
* **General Formula:** V(n) = $160,000 * (0.80)^(n-1)

**b) Year When Value is Less Than $100,000:**

We want to find the smallest integer *n* such that V(n) < $100,000.

1.  **Set up the inequality:**
    $160,000 * (0.80)^(n-1) < $100,000

2.  **Divide by 160,000:**
    (0.80)^(n-1) < 100,000 / 160,000
    (0.80)^(n-1) < 0.625

3.  **Take the logarithm (base 10 or natural log) of both sides:**
    log(0.80)^(n-1) < log(0.625)
    (n-1) * log(0.80) < log(0.625)

4.  **Divide by log(0.80). Note that log(0.80) is negative, so reverse the inequality sign:**
    n - 1 > log(0.625) / log(0.80)
    n - 1 > -0.2076 / -0.0969
    n - 1 > 2.1424

5.  **Add 1 to both sides:**
    n > 3.1424

6.  **Find the smallest integer:**
    Since *n* must be an integer, the smallest value of *n* that satisfies this inequality is *n* = 4.

**Answer:**

a) The formula for V is V(n) = $160,000 * (0.80)^(n-1)

b) The value of the bulldozer will be less than $100,000 in the 4th year.