To solve this problem, we need to find a linear function $V(t)$ that describes the value of the item as a function of time $t$. The given information is:

• At $t = 0$ (when the item was purchased), the value is $V(0) = 28$ dollars.
• At $t = 8$ (after 8 years), the value is $V(8) = 0.48$ dollars.

We are assuming the depreciation is linear, so the equation for $V(t)$ will be in the slope-intercept form:

$V(t) = mt + b$

where $m$ is the slope and $b$ is the initial value of the item (the y-intercept, which is the value when $t = 0$).

Step 1: Find the slope ($m$)

The slope $m$ of a linear function is given by the change in $V$ divided by the change in $t$:

$m = \frac{\Delta V}{\Delta t} = \frac{V(8) - V(0)}{8 - 0}$

Substitute the values:

$m = \frac{0.48 - 28}{8 - 0} = \frac{-27.52}{8} = -3.44$

So, the slope $m = -3.44$.

Step 2: Write the equation

We already know that $V(0) = 28$, so the y-intercept $b = 28$. Thus, the equation for $V(t)$ is:

$V(t) = -3.44t + 28$

Final Answer:

The value of the item as a function of time $t$ is:

$V(t) = -3.44t + 28$

This equation tells us that the item loses $3.44 per year.

Would you like more details on any part of this process?

Here are 5 related questions for further practice:

1. How much would the item be worth after 5 years?
2. At what time will the item have no value left?
3. How would the function change if the depreciation were exponential instead of linear?
4. What is the significance of the slope in this context?
5. Can you derive a similar function if the value decreases by 20% each year?

Tip: The slope in linear depreciation problems represents the rate of change of value per unit of time.