Modeling with Linear Functions
An office machine purchased new for $3600 loses $400 each year.
Assume the value continues to decrease the same amount each year. If f(x) represents the value of the office machine after x years, which linear function models the given situation?
A. f(x) = 400x + 3600f(x) = 400x + 3600
B. f(x) = −400x − 3600f(x) = −400x − 3600
C. f(x) = −400x + 3600f(x) = −400x + 3600
D. f(x) = 400x − 3600
How many years will it take for the office machine's value to equal zero?
Responses
A. 55
B. 66
C. 99
D. 12
To model the given situation, we need a linear function that represents the decrease in value of the office machine. Since the office machine loses $400 each year, the function must include a negative coefficient for the years.
Looking at the options:
A. f(x) = 400x + 3600 - This option has a positive coefficient for the years, which means the value of the office machine would increase over time, not decrease.
B. f(x) = −400x − 3600 - This option has a negative coefficient for the years, which represents the decrease in value. So this could be a possible linear function.
C. f(x) = −400x + 3600 - This option has a positive coefficient for the years, which means the value of the office machine would increase over time, not decrease.
D. f(x) = 400x − 3600 - This option has a positive coefficient for the years, which means the value of the office machine would increase over time, not decrease.
So, the linear function that models the given situation is:
B. f(x) = −400x − 3600.
To find out how many years it will take for the office machine's value to equal zero, we can set the linear function to zero and solve for x:
−400x − 3600 = 0
Adding 3600 to both sides:
−400x = 3600
Dividing both sides by -400:
x = -9
Since the number of years cannot be negative in this context, we discard the negative solution.
Therefore, it will take 9 years for the office machine's value to equal zero.
So the correct option is not listed among the given responses.