A machine is now worth ​$156 comma 700 and will be depreciated linearly over a 11​-year ​period,  at which time it will be worth ​$31 comma 190 as scrap. 
​(a) Find the rule of depreciation function f.
​(b) What is the domain of​ f?
​(c) What will the machine be worth in 7 ​years?
Question content area bottom
Part 1
​(a) Find the rule of depreciation function f.
​f(x)equals
  
156700 minus 11410 x ​(Do not include the​ $ symbol in your​ answer.)
Part 2
​(b) What is the domain of​ f?

The problem involves linear depreciation, so let's break down each part.

### Part 1: Finding the rule of the depreciation function $ f(x) $

You are given:
- Initial value of the machine: $ \$156,700 $
- Scrap value after 11 years: $ \$31,190 $

Linear depreciation means the value of the machine decreases at a constant rate. To determine the depreciation rate:

1. **Depreciation rate (slope)** is the change in value over time:
   \[
   \text{Depreciation rate} = \frac{\text{Initial value} - \text{Scrap value}}{\text{Time period}} = \frac{156700 - 31190}{11} = \frac{125510}{11} = 11410 \, \text{per year}
   \]

2. **Depreciation function**: The machine's value $ f(x) $ after $ x $ years is given by:
   \[
   f(x) = 156700 - 11410x
   \]
   
So, the rule of the depreciation function is:
\[
f(x) = 156700 - 11410x
\]

---

### Part 2: Domain of $ f(x) $

The domain of the depreciation function corresponds to the time period over which the machine's value is being calculated. Since the machine depreciates over 11 years, the domain is:
\[
\text{Domain of } f(x): \quad 0 \leq x \leq 11
\]
Where:
- $ x = 0 $ represents the time the machine is first purchased.
- $ x = 11 $ represents the time the machine reaches its scrap value.

---

### Part 3: Machine's worth after 7 years

To find the value of the machine after 7 years, substitute $ x = 7 $ into the depreciation function:

\[
f(7) = 156700 - 11410 \times 7 = 156700 - 79870 = 76830
\]

The machine will be worth $ \$76,830 $ after 7 years.

---

### Summary of Answers:
1. **Depreciation function**: $ f(x) = 156700 - 11410x $
2. **Domain**: $ 0 \leq x \leq 11 $
3. **Worth after 7 years**: $ \$76,830 $

Would you like more details on any of these calculations?

Here are 5 related questions:
1. How is the depreciation function derived?
2. How can depreciation be modeled in nonlinear situations?
3. What happens if the machine is sold before reaching its scrap value?
4. How is depreciation used in accounting?
5. How does salvage value affect depreciation?

**Tip**: The slope of a depreciation function is always negative because the value decreases over time.

A machine is now worth $156,700 and will be depreciated linearly over an 11-year period, at which time it will be worth $31,190 as scrap. (a) Find the rule of depreciation function f. (b) What is the domain of f? (c) What will the machine be worth in 7 years?

Find the linear depreciation function for a machine worth $156,700, depreciated over 11 years to a scrap value of $31,190. Learn how to calculate the depreciation rate, the domain of the function, and determine the machine's worth after 7 years. This problem is ideal for students in Grades 9-12 and covers algebraic concepts related to linear functions.

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