For tax purposes, a business depreciates its $240 comma 000 building using "straight-line depreciation" over 12 years by reducing its value by 1/12 of its original value each year. If we assume that the value of the building can be found or estimated at any point of time during this period, the value of the building can be modeled by the function yequals240 comma 000minus20 comma 000x, where x is the number of years. Complete parts (a) through (c).

Question

For tax​ purposes, a business depreciates its ​$240 comma 000 building using​ "straight-line depreciation" over 12 years by reducing its value by ​1/12 of its original value each year. If we assume that the value of the building can be found or estimated at any point of time during this​ period, the value of the building can be modeled by the function yequals240 comma 000minus20 comma 000​x, where x is the number of years. Complete parts ​(a) through ​(c).
What is the meaning of the​ x-intercept? Select the correct choice below and fill in the answer​ box(es) to complete your choice.
​(Type integers or​ decimals.)
A.
The value of the building decreases by ​$
  
enter your response here per year.
B.
At the beginning of the 
  
enter your response here ​years, the building is worth ​$
  
enter your response here.
C.
The value of the building is ​$
  
enter your response here at the end of the 
  
enter your response here years.
D.
The value of the building increases by ​$
  
enter your response here per year.

GPT-4o mini · Answer

To analyze the given function for straight-line depreciation of the building, we can start by examining the function itself:

\[ y = 240,000 - 20,000x \]

In this equation:

$y$ represents the value of the building after $x$ years.
The term $-20,000x$ indicates that the building depreciates by $20,000 each year.

Finding the x-intercept

The x-intercept occurs when $y = 0$. To find the x-intercept, we set the equation to zero and solve for $x$:

\[ 0 = 240,000 - 20,000x \]

Rearranging gives:

\[ 20,000x = 240,000 \]

Now, solving for $x$:

\[ x = \frac{240,000}{20,000} = 12 \]

Meaning of the x-intercept

The x-intercept of the function occurs at $x = 12$ years, which means the value of the building reaches $0 at the end of 12 years.

Selecting the Correct Choice

Now let's interpret the meaning of the x-intercept:

C. "The value of the building is $0 at the end of the 12 years."

Filling in the answer boxes:

C. The value of the building is $0 at the end of the 12 years.

Summary:

The x-intercept is 12 years.
This represents the point in time when the building's value has completely depreciated to $0.