To solve these questions, we will analyze the situation step by step.
Question 1: Modeling the Value of the Office Machine
The initial purchase price of the office machine is $3600, and it loses $400 in value each year. We define \( f(x) \) as the value of the office machine after \( x \) years.
The model should reflect the initial value minus the loss for each year. Thus, the equation can be constructed as follows:
- The initial value when \( x = 0 \) is $3600.
- Each year (for every increase of \( x \) by 1), the value decreases by $400.
Therefore, the mathematical representation of the value would be:
\[ f(x) = 3600 - 400x \]
This can also be rewritten in the slope-intercept form \( f(x) = mx + b \):
\[ f(x) = -400x + 3600 \]
Hence, the correct answer to Question 1 is option C: \( f(x) = -400x + 3600 \).
Question 2: Finding When the Value Equals Zero
To find out when the value of the office machine will equal zero, we need to set the function equal to zero:
\[ 0 = -400x + 3600 \]
Now, we solve for \( x \):
- Add \( 400x \) to both sides:
\[ 400x = 3600 \]
- Now, divide both sides by 400:
\[ x = \frac{3600}{400} \]
- Simplifying this gives:
\[ x = 9 \]
So, it will take 9 years for the office machine's value to reach zero. However, that option does not appear in your responses. Let’s check available responses:
- A) 55
- B) 66
- C) 99
- D) 12
While the exact answer still doesn't match any choice provided, if we were to consider the patterns of such problems, often the responses have typographical errors or omissions in the options.
Therefore, based on our calculation, 9 years is indeed the answer, but if you need to select an option from the given list, you may have to double-check if we miscalculated or if there was a typographical error in the question's context.
However, using the correct calculations above, we can conclude.
- First question answer: C \( f(x) = -400x + 3600 \)
- Second question calculated value: 9 years, which assumes a mistake in the provided options.
2 answers